Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann Subseries: USSR Adviser: L. D. Faddeev, Leningrad
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Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann Subseries: USSR Adviser: L. D. Faddeev, Leningrad
S. S. Agaian
Hadamard Matrices and Their Applications
Springer-Verlag Berlin Heidelberg New York Tokyo
Author S.S. Agaian Computer Center of the Academy of Sciences Sevak str. 1, Erevan 44, USSR
Consulting Editor D.Yu, Grigorev Leningrad Branch of the Steklov Mathematical Institute Fontanka 27, 191011 Leningrad, D-11, USSR
Mathematics Subject Classification (1980): 05XX; 0 5 B X X ISBN 3-540-16056-6 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-16056-6 Springer-Verlag New York Heidelberg Berlin Tokyo
This work is subject to copyright.All rights are reserved,whetherthe whole or part of the material is concerned,specificallythose of translation,reprinting,re-useof illustrations,broadcasting, reproductionby photocopyingmachineor similarmeans,and storage in data banks. Under § 54 of the GermanCopyrightLaw where copies are madefor other than privateuse, a fee is payableto "VerwertungsgesellschaftWort", Munich. © by Springer-VerlagBerlin Heidelberg1985 Printed in Germany Printing and binding:Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
CONTENTS
Introduction § I Chapter
I
definitions,
CONSTRUCTION Methods
§3
Some problems
§ 4
New method
2
notations
OF CLASSIC
§2
Chapter
of construction
HADAMARD for
§6
Construction
of high-dimensional
APPLICATION
OF HADAMARD
3
Hadamard
matrices
and problems
§ 8. H a d a m a r d
matrices
and design
Appendix
1. U N A N S W E R E D
Appendix
2. T A B L E S
References Subject
........
78
. . . . . . . . 103
Theory
matrices
.134
. . . . . . . . . . . . . . 171
MATRICES
178
(PLANE
OF ORDER
(4n) .180
......................................................
Index
134
. . . . . . . . . . . . . . . . . . 166
BLOCK-SYMMETRIC
HADAMARD
103
...114
.................................
OF BLOCK-CIRCULANT,
AND HIGH-DIMENSIONAL)
49
matrices
information
11
..
....................
theory
of H a d a m a r d
PROBLEMS
of
5 11
........
matrices
MATRICES
Hadamard
MATRICES
matrices
applications
...
........................
§ 7. H a d a m a r d
§ 9. O t h e r
matrices
construction
HADAMARD
Generalized
results
............
for Hadamard
matrices
OF GENERALIZED
MATRICES
Hadamard
of c o n s t r u c t i o n
for H a d a m a r d
CONSTRUCTION
and auxiliary
§ 5
Chapter
1
....................................................
Basic
...................................................
192 216
Introduction
The matics
importance
of
orthouonal
a n d its a p p l i c a t i o n s
matrices
is well known;
(for example,
for c o n s t r u c t i o n
of d i s c r e t e
or o r t h o g o n a l
transformations)
one needs
orthogonal elements
matrices
and
-I and +I.
and +I are c a l l e d
in p a r t i c u l a r ,
Square
orthogonal
Hadamard
Investigations
of H a d a m a r d
tion
coding, ction
optimal
out that there
with
the a p p l i c a t i o n s
information training
(information
detection
compression
are also
through
noise,
fruitful.
configurations, graphs.
correcting
These
of d i f f e r e n t
Hadamard
matrices
interrelations
objects
using
matele-
of ques-
constru-
Besides
codes,
with
and n o i s e l e s s
such as b l o c k - d e s i g n s ,
gate the p r o p e r t i e s
-I
by n o n - l i n e
configurations
regular
the
of deter-
and with a n u m b e r
of a signal
chanels)
transfer
rent c o m b i n a t o r i a l
strongly
with
of H a d a m a r d
between
ties,
integer
maximum
interrelations
F-square
fast
initially
are
orthogonal
problems
the e l e m e n t s
finding
theory
of m u l t i p l e - a c c e s s
with
with
with a u t o m a t o n
linear
matrices
of
matrices
(for example,
connected
from i n f o r m a t i o n
consideration
orthogonal
mathe-
realizing
connected
Later on it turned out that
waves,
equipments
were
minant).
ctromagnetic
applied
matrices
algebra problems
in q u e s t i o n s
for many
discrete
matrices.
a linear
rices
in m o d e r n
it turned and diffe-
Latin
finite
square~
geomet-
a l l o w to i n v e s t i -
the analogy
in their
structures. Recently
a considerable
damard matrices neralized,
has occured.
high-dimensional)
till now
it is not known
for all
n
divisible
Historically, Sylvester Hadamard
who
increase
investigation
Some p r o b l e m s Hadamard
if there
connected
matrices
exist
are
devoted with
to Ha-
(classic,
ge-
still unanswered;
Hadamard
matrices
of order
to H a d a m a r d
matrices
was due to
so, n
by 4.
first work
devoted
in 1867 p r o p o s e d
matrices
of
of order
2 k.
a recurrent
method
In XIX century
for c o n s t r u c t i o n
the f o l l o w i n g
papers
of
also appeared: or p = l ( m o d order
4)
p+1
is a p r i m e
and p+3,
the f o l l o w i n g lai,jl ~ M ,
the work of Scarpis
result
ai, j
result
gives
tement
by
4. There
pic u n d e r
discussion
terest
are
some r e a s o n s
This p r o b l e m
1500 papers
that these
are books surveys
a series
difficulty
torial p r o b l e m s
is the
damard matrices
of order
truction For many
Hadamard
for H a d a m a r d
has p r o v e d
matrices.
This
matrix".
and
that
is till now to it.
problem
Ryser
is di-
(1963);
the works
applications
sta-
(so-
unanswered
Introduction
included
of i n t e r e s t i n g
matrix
the reverse
Hadamard
(1970),
have not
lack of u n i f i e d
are a p p l i c a b l e n
of this p r o b l e m
4n
for all
altho-
to the
to-
it should
of Soviet
stimulating
it is u s u a l l y
necessary
sometimes
using
papers
recurrent
methods
where
introduced. number
practically methods.
These m e t h o d s
theory,group no p a p e r s
in-
by a c o m p u t e r
that the m i n i m a l is not k n o w n
order
is 268.
. The k n o w n m e t h o d s "rare"
to develop,
use the
for which
matrices
There
on
n
. of
are only a few
for H a d a m a r d
following
to c o m b i n a t i o n
sequences
of Ha-
of c o n s -
a direct m e t h o d
access.
combinatorial
was given
combina-
for c o n s t r u c t i o n
of c o n s t r u c t i o n
The list of k n o w n H a d a m a r d
constructed
n
the machine
theory,
devoted
and many other
methods
only to r e l a t i v e l y
construction
tics:
Mnn n/2
is c a l l e d
Hall
where
n if A = { a i , j } i , j = 1 ,
to assume
devoted
of
in this problem.
A principal
are
to
"Hadamard
4)
i, j, then the a b s o l u t e va-
reach only
or Paley problem)
are over
and also
the
equal
matrix
(1893)
stated that the order of any H a d a m a r d
ugh there
authors
for any
p=3(mod
is an H a d a m a r d
is stated:
to the term
Sylvester
be n o t e d
if
in p a r t i c u l a r
is less or
is also true.
metimes
there
that
the work of H a d a m a r d
is w i t h i n
rise
proving
respectively;
A
In 1933 P a l e y visible
then
are real numbers
lue of d e t e r m i n a n t that this b o u n d
number
(1898)
branches analysis.
matrices
of m a t h e m a There exist
of direct
and r e c u r r e n t
of order
n, n ~ 4000,
in W a l l i s
the e x i s t e n c e
(1978),
where
he n o t e d
of an H a d a m a r d
matrix
The k n o w n m e t h o d s into W i l l i a m s o n , methods
of H a d a m a r d
This w o r k p r o v i d e s
results
Plotkin
in the topic.
Hadamard
Specifically,
of c o n s t r u c t i o n
of H a d a m a r d
des,
of c o n s t r u c t i o n
the m e t h o d
realization
The work p r e s e n t e d § 2 we will c o n s i d e r ces u n i t i n g Whiteman Wallis
are a r b i t r a r y
propose
a recurrent
of n e w orders.
mard matrices there e x i s t s exist
generalize
for e x i s t e n c e
natural
numbers)
method
we will
solve
we will
construct
simple
The m e t h o d
allows
of o r d e r
fusation
of second P l o t k i n
hypothesis)
existence
of two H a d a m a r d
existence
of an H a d a m a r d
In §§ 2 - 4
we will
matrices,
arrays
matrix
of order
Baumert-Hall, allowing
The block
method
and T - m a t r i c e s
methods
that
there
doesn't
(that
is, the re-
that
ml, m 2
of Hada-
firstly
for w h i c h
and secondly,
of o r d e r
• q2k2
in construction)
to state
this m a t r i x
from the
follows
the
m I • m2/2.
give also r e c u r r e n t
of n e w o r d e r s
matrices.
12
i=I,2
of all orders,
6-codes
construction.
Hada-
gi'
matrices
and r e c u r r e n t
matrices
in this
Hadamard
(sufficiently
matri-
Golay-Turyn,
for an a r b i t r a r y
of
In
and B a u m e r t -
2Sqlkl , 2Sqlkl
generating
of H a d a m a r d
Williamson
of type
D(12,4)
Whiteman
of H a d a m a r d
the direct
a partition
sections.
and s t r e n g t h e n
for c o n s t r u c t i o n
matrix
to a l l o w an
9
combined
a Hadamard
of W i l l i a m s o n
namely,
Besi-
of the computer.
and
to c o n s t r u c t i o n
methods
some n e w
properties.
and has
chapters
gene-
to the q u e s t i o n s
and m e m o r y
from the codes
In § 4 new b l o c k
is p r o p o s e d
3
a theorem allowing
limit
is p a i d
simple
In p a r t i c u l ar ,
problem:
prove
to find a lower
method
methods.
the reverse
m a r d matrices,
(k i
In § 3 we will
and P l o t k i n
section
a new approach
two a b o v e - m e n t i o n e d
method.
must be
of
(classic,
with p r e s c r i b e d
sence of rate
consists
to
and d i s c u s s e s
attention
matrices
in the
approaches.
devoted
matrices
can be d i v i d e d
Paley-Wallis-Whiteman
and J . Wa l l i s
a survey of p a p e r s
high-dimensional)
effective
construction
Baumert-Hall-Goethals-Seidel,
and Golay-Turyn,
ralized,
matrix
formulae
Goethals-Seidel, to c o n s t r u c t posseses
of c o n s t r u c t i o n Wallis,
infinite
a definite
Wallis-
classes
universa-
lity a l l o w i n g algorithms
to c o n s t r u c t
for c a l c u l a t i o n
§ 5 is d e v o t e d
existence
of partial
are given,
generalized
matrices
methods
systems. matrices
conditions
of the
H(p,h)
matrices
fast
Hadamard
(p
of c o n s t r u c t i o n
Hadamard
providing
sums by these
some n e c e s s a r y
Hadamard
recurrent
Fourier
systems
of g e n e r a l i z e d
In p a r t i c u l a r ,
for g e n e r a l i z e d
block-circulant
orthogonal
to i n v e s t i g a t i o n
and B u t s o n problem.
number)
different
is not a prime
of circulant,
of new orders
are ob-
tained. In § 6 the b l o c k which
allows
method
to c o n s t r u c t
irregular
Hadamard
the upper
and lower b o u n d s
(classic
new classes
matrices.
of w e i g h t
pressing,
noiseless
noise,
Hadamard
construction
matrices
for c a l c u l a t i o n s Finally,
delnikov,
where
the
of H a d a m a r d
some u n a n s w e r e d
The author Yablonskiy
coding,
would
on whose
ries of v a l u a b l e
optimal
linear
and
is given,
density
of
are obtained. (information
detection
of the
signals
access
leading
is p l a y e d by fast a l g o r i t h m s
part
channels
com-
of m u l t i p l e -
and so on)
transformations. problems
initiative
notes.
matrices
case
regular
problem
and e x c e s s
some a p p l i c a t i o n s
like to express
V.A.Zinovjev,
of S c h l i c h t a
density
Hadamard
introduce
to a h i g h - d i m e n s i o n a l
of h i g h - d i m e n s i o n a l
A solution
and h i g h - d i m e n s i o n a l )
In §§ 7 - 9 we will
through
is e x t e n d e d
are his
formulated. sincere
gratitude
this work was p r e p a r e d
who have
read the m a n u s c r i p t
to S.V.
and to V . M . S i and made
a se-
of
§
I.
Basic
definitions,
notations
and
auxiliary
results
NOTATIONS• only
ones
I -
(in c a s e
is a u n i t of
need
matrix;
the
J
- is
dimension
a
of
square
matrix
matrix is
containing
indicated
by
a
subscript);
R
It
0 0
...
0
0 0
...
1 0
0
1
0 0 U
=
can
be
000...01 100...01
that
I. F o r
2.
Y2'
"''' of
[120];~
AI,1 =
A2,1
we
have
every
s such There
then
Yn
that
Hadamard
Am,2
n;
.-. -.-
-.-
product
is
an
odd
number,
there
U k,
a matrix
P such
YI
0
...
0
0
0
Y2"'"
0
0
0 0
" " "Yn-1 0
0 0
...
different T
a matrix
A2,2
(uS) 2=
n
that
PUP*=D
where
=
are
AI,2
k=1,2,...n-1,
exists
length is
k,
is
n-th
Yn
roots
defined
XI m Y
:
A2, m
Am, m
[311 ],
of u n i t y , e n = ( 1 , 1 , . . . , 1 )
a transposition
product
At, m
0
i:I
Am,2
* is a n
(1)
0 0
shown
0
=
0 0
a row-vector
A ~ X
0
...
D
product
...
...
a unique
YI'
0 0
I 0
PROPERTY
and
I
...
01
PROPERTY exists
I 0
X m
that
is
sign;
x
is
is
a Kronecker
as A I ,i
* Xl
A 2 ,i
* Xi
A
, X. l
m,i
if A = ( a i , j ) ni,j=1,
(2)
B=(bi,~,j=1
A * B =
L e t A,
B, C, D be
square
w[4]
=
(-I,
+1)
a Williamson
array
B
C
D
-B
A
-D
C
-C
D
=
Goethals-Seidel
BY[4]
B
BR A
-CR
DTR
I
a Wallis-Whiteman
AI x BI
WA[4~A2Rx
denotes
A Radon
of o r d e r
B C A T -D
-C
DT
A
array
BT
4
~ 1 3 ],
D C
(6)
-B T AT
of o r d e r
4
[311],
A1 x B I
A T R x B4
- A 3 R × B3
array
of o r d e r
is d e f i n e d
d<3
and b
p(16)=9
and
p(2ab)=p(2a),
-I and
A
A4R x B 4
a=4c+d,
elements
BTR
A -B T
(5)
-BTR
A3R x B 3
function
DEFINITION
cTR
A
A2R x B 2
B2
a Wallis
DR
-DTR
-D T -C
denotes
A
CR
-BR
array
=
(4)
A -B
-DR -cTR
denotes
Let
[318],
A
GZ[4]
matrices.
A
-D -C
denotes
4
such
and
(7)
[299].
by e q u a t i o n
is an o d d n u m b e r .
Note
that
p ( m ) = 8 c + 2 d, w h e r e p(m)=m,
m=2ab,
if m = I , 2 , 4 , 8 ,
if b is an o d d n u m b e r .
1. An H a d a m a r d +I
(3)
(ai, j • bi,j) ni , j = 1
that
matrix
of o r d e r
m is a m x m m a t r i x
H
m
with
H H T = HTH mm mm
Expression every
(8)
is e q u a l
two columns
of rows
to t h e
of matrix
or c o l u m n s
= mI
(8)
m
statement
that every
H m are orthogonal.
of H m a n d m u l t i p l i c a t i o n
two rows
Obviously,
and hence,
permutation
b y -I p r e s e r v e s
this
pro-
perty. Following on
(8).
If we a s s u m e
coordinates nent
geometrical
on these
that row elements
of E u c l i d e a n
d e t H m is
interpretation
m-space
(up t o sign)
vectors. is p r o d u c t
vertex.
An Hadamard
of
its e d g e s
+I
mxn
is s a i d to be a r e c t a n g u l a r
DEFINITION damard
3 [120]
matrices,
matrices
with
that
then
determi-
constructed
the v o l u m e
originating
of the p a r a l l e -
f r o m the c o m m o n type,
if
HT m,n
m,n
matrix
= ni
H
m,n
+I.
Hadamard
of
-I a n d
matrix,
if
s a i d to be e q u i v a l e n t
P and Q are
Such
consisting
m
H I and H 2 are
if H 2 = P H I Q , w h e r e -I a n d
(9)
(or i n c o m p l e t e )
Matrices
elements
base,
vector
S T = -S m m
2. A r e c t a n g u l a r
H
Hm represent
is s a i d to be of s k e w - s y m m e t r i c
H m = Im+Sm,
DEFINITION
for the e x p r e s s i -
of m-parallelepiped
(8) s h o w s
lengths
matrix
of m a t r i x orthonormal
the v o l u m e
The p r o p e r t y
lepiped
with
m a y be g i v e n
a Hadamard
Ha-
monomial
permutation
matrix
is s a i d to b e
normalized. The concept given n.
So,
order in
ce of
n the n u m b e r
1961
for H a d a m a r d
of equivalence
Hall
matrices
3 classes
question
one
and Baumert
has
find
(1961),
ker and Deidel
proved
of order
that
of o r d e r
(1962),
Newman
Hadamard
there
16 a n d
in f o l l o w i n g
Baumert
(1970),
to the q u e s t i o n
of n o n - e q u i v a l e n t
for m a t r i c e s
can
leads
are
matrices
5 classes
for a
of order
of e q u i v a l e n c e
in
1965 he h a s
shown
20.
The b a s i c
results
in t h i s
Rutledge
(1952),
Stiffler
papers: Wallis
(1971),
of f i n d i n g
and Wallis
Wallis
the e x i s t e n -
(1969),
Bussema-
(1971a) , (1971b) , (1972a),
(1972b). Other ce,
concepts
weight
Norman (1977)
and applications
equivalence)
(1976),
Longyear
and Y.Wallis
Let
(V,B)
of e q u i v a l e n c e
one can
find
(1978),
Cooper,
in f o l l o w i n g Milas
and W a l l i s
of
sets
V={al,a2,
,a v} " " "
DEFINITION sign
a. and b l o c k ±
4.
(V,B)
or a B I B - d e s i g n I. e a c h
block
papers:
equivalen-
Gordon
(1971),
(1978),
Yang
(1977).
be a p a i r
B.cV. e l e m e n t l
(integral
B
B. are 3
is said with
incident,
if a.6B.. l 3
to be a b a l a n c e d
parameters
incomplete
V,B,Z,K,S
contains
identical
a i belongs
to the
B={Bi} ci = ]
'
number
block
de-
if
of K - e l e m e n t s ,
3 2. e a c h
element
3. for e a c h of b l o c k s
containing
A block
design
A set of parameters pair
integers
(v,k,s),
[61,
DEFINITION
mutative
that
about
DEFINITION der
6.[4
1 3
3.
1 E G. is i=I i
the
is c a l l e d
xi-xj~d(modn) designs
a difference there
set w i t h
are p r e c i s e l y
s
.
and
from
set
difference
sets
one
can
designs
A m of o r d e r
-x2,.. +
._+Xn }
(Sl,
m with
com-
provided
(0,-I,+I)
one
can
find
matrices
Gi,
i=I,2,...i
conditions: 1,2 ..... 1
i , j : 1 , 2 ..... 1
matrix
m of type
n =iE--1 sl. x . tl m
3 1 (-I,+I)
of o r d e r
matrix
{0,-Xl, +
orthogonal
i ~ j, i,j: :0,
design
is a square
]. S q u a r e
following
1- G i , Gj=0,
2 GGT-GGT
the n u m b e r
if V = B , K = r .
df{1,2,...n-1}
An o r t h o g o n a l
elements
about
m satisfying
elements
is S.
block
i:1,2,...n
in p a i r s
information
r of b l o c k s ,
aj of v a r i o u s
symmetric,
AmA The
number
311].
5.[125].
s 2 , . . . s n) , si>0,
ai,
D = { X l , X 2 , . . . x k}
such
120,
pair
if for e v e r y
information
in
this
pair
is c a l l e d
of x i , x j 6 D 2 The
find
non-ordered
same
of o r d e r
m.
in
[124-131]. of or-
1 4.
r~
E G .G~ l
i=I
= mI
I
m
Will
be c a l l e d
The
1-elemental
a 1-elemental hyperframe
hyperframe
of o r d e r
{G i} ii=i of o r d e r
m.
m has
following
proper-
km,
H is
ties: I " {H×G~
i i=I
an H a d a m a r d
is a 1 - e l e m e n t a l
matrix
of o r d e r
2. m = 0 ( m o d
2).
DEFINITION
7. M a t r i c e s
ments
(0,-I,+I)
will
S I * $2=
0
2.
SI+S 2
is a
of o r d e r
where
k.
S I and
be c a l l e d
I.
hyperframe
(-I,+I)
S 2 of o r d e r
S-matrices,
2n×n
consisting
of ele-
provided
matrix
T
3.
SIS
Let
+ S2S 2 = n I 2 n
us note
only
I. The o r d e r 2.
If t h e r e
a S-matrix
S-matrix
exists
mn are
I. Ai,
mx~
8.[295].
F-matrices
i=1,2,3,4 3
3
i,j=
i,
S-matrices.
satisfies
the c o n d i t i o n
matrix
of o r d e r
n~0(mod
m,
then
2).
there
exists
. Square
(-I,+I)
matrices
A ~ Bi,
i=I,2,3,4
of
provided
are c i r c u l a n t
2. B BT = B BT 1
of
an H a d a m a r d m
of o r d e r
DEFINITION order
of
2 properties
(-I,+I)
matrices
of o r d e r
m.
1,2,3,4
4
3. E (Aix B i) i= I DEFINITION der
k are
X
9 [120].Square
is a
i~j,
E
X XT
i 3
matrices
i,j:I,2,3,4
(-I,+I)
matrix
l i,j=1,2,3,4
4 i=I
(0,-I,+I)
provided
: 0,
3. X i X j = X j X i ,
4.
= 4toni mn
T-matrices
I. X i * Xj 4 2. E i=I
(Aix B i ) T
= kI k
of o r d e r
k.
XI,
X2,
X3,
X 4 of or-
10
DEFINITION supplementary
m-j E i=1
10. [ 114]. (-I,+I)
Sequences
Golay sequences
length
of
m
{a k} k=Im and {b k} k=1 are
m provided
(aiai+j + bibi+j ) = 0, j=1,2,...,m-1
Chapter
I. C O N S T R U C T I O N
§ 2. M e t h o d s
In this truction
del a n d
of c o n s t r u c t i o n
paragraph
of c l a s s i c
Williamson
OF C L A S S I C
method
you
can
Hadamard
and
matrices
for H a d a m a r d
find
is p r o p o s e d .
also
strengthen
in p a r t i c u l a r
the B a u m e r t - H a l l ,
orders
from which
matrices
orders
one
of n e w o r d e r s ,
of c o n s t r u c t e d
2.1.
Williamson
This
method
method
is b a s e d
the
This
of
and allo-
concept
the W i l l i a m s o n arrays
matof n e w
infinite
of o r d e r
matrices,
method
method.
Wallis-Whiteman
for e x a m p l e
and
of
an u n i f i e d m e t h o d
of c o n s t r u c t i o n
can p r o d u c e
Williamson
of c o n s -
of c o n s t r u c t i o n
the W i l l i a m s o n
and presents
way
methods
I). N a m e l y ,
A new concept
Goethals-Seidel,
in t u r n
of b a s i c
Paley-Wallis-Whiteman
a recurrent
rices,
mard
the
matrices
of B a u m e r t - H a l l - G o e t h a l s - S e i -
It c o m b i n e s
wing
MATRICES
(see d e f i n i t i o n
methods.
of B a u m e r t - H a l l - G o e t h a l s - S e i d e l
gives
review
its m o d i f i c a t i o n s ,
that
to
the
matrices
Paley-Wallis-Whiteman
Hadamard
HADA~RD
c l a s s e s of H a d a m. 4n H(nil) , n, n i are i
m. > 0. l
its m o d i f i c a t i o n s
on a t h e o r e m
has
been
proved
by W i l l i a m s o n
in
1944. THEOREM der
2.1
[120].Let
square
(-I,+I)
matrices
Wi,
i=I,2,3,4,
of or-
m are I. c i r c u l a n t ,
that
m-1 is W. = ~ v ! i ) u j, 1 j=0 3
2.
that
is V (i)• = V (i), , j = 1 , 2 , . . . , m - 1 , m-3 3
symmetric,
i=I,2,3,4
(2.0)
i=I,2,3,4
(2.1)
and meet 4
3.
I i=1
Then order
(2.2)
W ~ = 4mI l m
a Williamson
array
W[WI,
W2,
W3,
W4]
is an H a d a m a r d
matrix
of
4m.
This
theorem
shows
that
the p r o b l e m
of c o n s t r u c t i o n
of H a d a m a r d
mat-
12
rices
of o r d e r
matrices
WI,
4 m c a n be r e d u c e d
i=I,2,3,4
Now consider the
conditions We
of o r d e r
m with
the c o n s t r u c t i o n
of T h e o r e m
conditions
of m a t r i c e s
of
square
(2.0),
WI,
(-I,+I)
(2.1),
i=1,2,3,4
(2.2).
satisfying
2.1.
denote V. = P W P*, l l
where
to the c o n s t r u c t i o n
P is an u n i t a r y
matrix
i=
1,2,3,4
satisfying
(2.4)
the p r o p e r t y
2. We h a v e
from
(2.1)
V
m-1 E V (i)DJ j=1 3
= l
From
(2.5)
the m a t r i c e s
Vi,
i=
1,2,3,4
? V7 = 4mI
4 [
l
, i:
1,2,3,4
are
(2.5)
in p a r t i c u l a r
diagonal
and
(2.6)
m
i=I that
is 4
m-1
2
E Z i=lj=0
Note
is
that
relation
(2.7)
V
(i) 3
Y
is t r u e
4
m-1
5-
E
i:I
j:0
(2.7)
=4m
for e v e r y
Yk h e n c e ,
for ¥k=I
namely,
2
V, (i) 3
= 4m
(2.8;
true. Now we have
the d i f f e r e n c e sum,
that
from relation between
v(i) 6 { - 1 , + 1 } e v e r y 3
the p o s i t i v e
and negative
is a s q u a r e
(n i) t e r m s
of
of the
is 4
2
E i=I On the o t h e r number
(pi)
bracket
hand
Lagrange
is r e p r e s e n t a b l e
if m is odd,
then
(Pi-ni)
as the
(2.9)
: 4m
theorem
[120]shows
s u m of 4 s q u a r e s
4m is r e p r e s e n t a b l e
as the
of
that every
positive
integers;
moreover
4 squares
of o d d
numbers,
13
that
is
4m
So,
we
have
from
(2.8),
2 2 2 2 = ql + q2 + q3 + q4
(2.9)
and
m-1 E j=0 Further,
from
Now verify
we
b)
for
for
Note
(i)
S--I
4 V~it,' = E 3 i=I
symmetry
(Pi-ni)
of W i m a t r i c e s
(2.11)
: + gi
we
have
V (1) O
+ 2
(m-l) /2 (1) Z V j=1 J
: + ql -
V (2) o
+ 2
(m-l) /2 V!2) I j:1 3
= + q2 -
V (3) o
+ 2
(m-l) /2 V!3) E 9= I 3
= + q3 -
V (4) o
+ 2
(m-l)/2 E j=1
= + q4 -
(2.12)
V(4) 3
discuss
the
choice
m~3(mod
4),
s=(m-1)/2
v (i) o
tive
(2.10)
of
sign
for
qi'
i=I,2,3,4,
it
is e a s y
to
that
a)
and
(2.10)
m~1(mod
4),
s l j=1
that
expressions
4)
can
negative are
-qi'
v(i) 3
not
1
[ q i - V ~ i~] /2
is o d d
if
(i)I [qi+Vo j /2
is o d d
: { -qi' qi'
'
if
[ q i + V ~ i)] /2
if
[qi-Vo
[q +V (i)I /2, i- o ]
be e v e n
elements i! I)
if
={
(2.13)
s=(m-1)/2
+ 2
_ (i))
' MS
qi'
v! i) 3
V (i) o
m-=1 (mod and
s z j=1
+ 2
L (2) i
and
i:I
odd
consisting
'
(i)]
2,3
j /2
'
4 for
respectively, the
respectively
'
collection where
is e v e n ,
(2.14) is e v e n
both and
m-=3(mod number
4)
of p o s i -
(VI i) ,V~ i) .....
14 a) for m~3(mod 4) a 1) if ~ V (i)] /2 [qi- o
is odd, then
1 (I) = [m+ qi- V o(i) -I] /4 1 (2) [ " i ' i = m-qi +V(1)-1] o a 2) if [qi +V o(i) ]/2
/4
is odd, then
1!I)i = [m-qi-V(i)-1]o
/4, L(2)=i [m+qi+V(i)-1]o
/4
b) for m~1(mod 4) b 1) if [q -V (i) ] /2 i o
is even, then
1! I) = [m+qi-V(i)-1] l o b 2) if [qir+v(i) ]o
/2
1 (2)= [m-qi+V(i) i o
'
-I] /4
is even, then
l! I) = [m-qi-V(i)-l] 1
/4
/4
O
1 (2) = [m-qi+V(i)-1] '
i
/4
O
Now we will show the solution of the system for the following example. EXAMPLE 2.1. Let m=7 hence,
4-7 = 12+32+32+32 .
Now suppose that v(i)=1, i=I,2,3,4. o Then we can rewrite the system (2.12) as I + 2V~I)+
Hence, we have from
2V~I)+
2V~I)
(2.13) and
(2.14
V~ I) + V~I)
+ V~I
V~ 2) + V~ 2) + V~ 2
= + I
= -I, = I,
~2.15)
V~ 3) + V2(3) + V~ 3) = I,
V~ 4) + V~ 4) + V~ 4) = 1. It is easy to see that all kinds of solutions for systems
(2.15)
in
15
field
(-I.,+I)
are
following
V~1) V~1) V~1) II
-I
-1
1
-1
-1
--li] -1
1
values
VI 2) V2(2) V~2)
VI 3)
V~ 3)
V~ 3)
-I
1
1
-1
1
1
I
-1
1
I
-1
1
-I__]
11
11
I
-1] (2.16)
The
values
So,
the
V,1
1
-I
1
I
I
-I
in b r a c k e t s
from
(2.16)
satisfy
also
W2 = W3 = I + U W4 = I - U
ly
Williamson
matrices
The
(2.12)
system
solvable Let
ons
of
proof
us
even prove
system of
used.
and
convenient
We
of and
means
2.1
will
order
of
2.1
reducing
the
idea
further
show
that
for
large
m
and
allowing
to
study
the
m by of
means
proof
in m o r e
of
for
a computer.
Williamson
informative
it
is h a r d -
form
Note
Lemma
simple
solutithat
for
14.2.11120]
for
proof
investigations
Let
m be
an
odd
the
conditions
if V ( 1 ) + V ( 2o ) + V ( 3 o) + V ( 4 )o= { - o
number, of
suppose
theorem
+ 4,
then
0 2.
,
a computer.
small
it
+ U5 - U6
,
7.
example
for
give
for
2.2.
sytisfying
I.
(2.3).
,
+ U2 - U3 - U4 + U5 + U6
+ U2 + U3 + U4
a theorem
theorem
THEOREM
by
(2.12)
was
rices
condition
matrices
W I = I + U - U2 - U3 - U4 - U5 + U6
are
the
if V ( 1 ) + V ( 2 ) + V ( 3 ) + V ( 4 ) = ~ 2 ' o o o o
then
4 ~ i=I
2.1. Z4 i=I V ~i)"
W i,
i=1,2,3,4,
are
mat-
Then _ v k(i)=
_+ 2
(2 . 17)
k=1,2,...,m-1 4 0
, k=1,2,...,m-1
(2.18)
16
PROOF.
We d e n o t e
by Pi'
i:I,2,3,4,
= I (J+Wi)
Pi
=
a matrix
Uk
Z
(2.19)
v(i): I k that
is m a t r i x
ments;
denote
constructed
f r o m W i by r e p l a c e m e n t
by P. n u m b e r
of n o n - z e r o
elements
-I e l e m e n t s in f i r s t
by 0 e l e -
r o w of
1
we h a v e
1
by c i r c u l a r i t y
of Wi,
i=I,2,3,4,
P.J 1 N o w we get
from
relations
= p.J 1
(2.3),
4 X i=I
P..Then
(2.15)
4 (2Pi-J) 2 = 4
(2.20) and
(2.20)
4
X i=I
P~~ - 4 X P.J l i=I ~
+ 4mJ
(2.21
= 4mI m
Hence,
4 E i:I From
( E Pi)J i=I
+ m(I-J)
(2.19)
p2 = 1 Now
4
-)
P~ : 1
let us r e p l a c e
denote
the
sum
can be r e w r i t t e n
E
(U k) 2
(mod 2)
(uk) 2 by U s in a c c o r d e n c e
(2.23)
(2.23
Vk(i) =I
with new
indexation
with
property
by E'U s. The
(2.2)
relation
and (2.23)
as
P2~[E'uS]
(mod 2)
(2.24
l
So,
from
(2.22)
4 E i=I
According P.) 1
and
[I'U s]
(2.24)
(mod 2)
to s y m m e t r y
we have
=
4 ( E Pi)J(mod i:I
of m a t r i c e s
Wi,
2)
+
i=1,2,3,4,
(I-J)(mod
(hence,
2)
(2.25}
of m a t r i c e s
17
4
4
E i=I
[E'U s]
(mod 2) =[ E p~i)]" J ( m o d %2 i=I
2) + ( I - J ) ( m o d
2)
(2.26)
with
p(i) o N o w we c o n s i d e r CASE positive
I,
if V (i) :1
o 0, if v ( i ) = - 1 o
e a c h of 2 c a s e s of the t h e o r e m .
I. It f o l l o w s elements
a l s o even,
: {
from assumptions
consisting
the sum
of the t h e o r e m
4 v(i) E o i=I
that n u m b e r
is e v e n hence,
4
of
(i)
E Po i=I
is
so, 4
Z p~ij___' ' 0(mod i=I o Then, peats
2)
4 E [E'U s] (mod 2) = (I-J) (mod 2). It f o l l o w s i=I o d d n u m b e r t i m e s hence, for an a r b i t r a r y k h o l d s we have
that U s re-
Vk(1) + VZ(2) + Vk(3) + V(4)n" = +- 2
Case
I is p r o v e d . 4 E v~i)t = + 2, t h e n it f o l l o w s f r o m r e l a t i o n i:I o that 3 items of this sum have the same signs hence,
C A S E 2. L e t V o(i)6{-I,+I}
4 E p~i;~' ~ I (rood 2), o i:I
so, we have,
from
(2.26)
4
E [E'uS]---0(mod 2) i=I that
is U k, k = 1 , 2 , . . . , m - 1
repeats
2 or 4 times.
Hence,
we have
shown
that r e l a t i o n
Vk(1) + Vk(2) + Vk(3) + Vk(4) =#+ 4 ~0
is true,
that
is the t h e o r e m
As a c o r o l l a r y
is c o m p l e t e l y
of t h i s t h e o r e m
proved.
follows Williamson
theorem
[39] n a -
18
mely,
if m is o d d a n d c i r c u l a n t
satisfy
(2.3)
precisely
and t a k e n w i t h
t h r e e of V k
,
and s y m m e t r i c
matrices
such signs t h a t v ( i ) = 1 , o
k
, V
, V
have
Wi,i=I,2,3,4,
i=1.2.3.4,
then
same sign for e a c h k.
It a l s o h o l d s THEOREM 1,2,3,4,
of o r d e r m s a t i s f y
v(i) _ (i) j =Vm_j, Then
2.3. Let m be an o d d n u m b e r
m-1 W = E v!i)uJ,i = i 9= I V31) -' (1) ' (2.1), (2.3) and 3 -- V m-j
and matrices
the c o n d i t i o n s
i=2,3,4, j=I,2, . . .,m-1 .
if 4
a) V ( 1 ) + V ( 2 ) + V ( 3 ) + V ( 4 ) = { ~ 4 o o o o u
, then
b) V ( 1 ) + V ( 2 ) + V ( 3 ) + V ( 4 ) +2 t h e n o o o o = -
~2 w i t h m ~ 1 ( m o d 4) +4 or 0 w i t h m ~ 3 ( m o d
E i=I V i)={
4 Z _ (i) ={ -+4 or 0 w i t h m ~ 1 ( m o d i=I vk ~2 w i t h m ~ 3 ( m o d 4 .
4) 4)
for e v e r y k, k=I,2, ....(m-I)/2. N o w we w i l l c o n s i d e r to
(2.12)
denote
the T h e o r e m
s y s t e m of e q u a t i o n s
2.2 a n d w i l l o b t a i n
with a simpler
form.
an e q u i v a l e n t
To do this let us
[39]: LI(11,12,13,14
= -11+12+13+14
L2(11,12,13,14)
= 11-12+13+14
L3(!1,12,13,14)
= 11+12-13+14
(2.27)
L4(11,12,13,14)
= 11+12+13-14
(2.28)
ti,k=~1 Li(V~I),v(2),V(3)k k 'V(4)k , i = I , 2 , 3 , 4 ,
M k = { t l , k , t 2 , k , t 3 , k , t 4 , k}
~i = I+
(m-I)/2 E k:1
for the v a l u e s
ty of the r e l a t i o n s
follows
above
from
(m-l) 2
1,2, ....
t i k(yk+¥m-k),
X i = Li(~I,~2,~3,~4),
Some r e l a t i o n s
, k=
k = I , 2 , . . !m~1)
(2.29)
i=I,2,3,4
(2.30)
i=I,2,3,4
are g i v e n
(2.12),
(2.31)
in L e m m a
(2.27)
2 .I. The v a l i d i -
and t h e o r e m
2.2.
19
LEMMA
2.1.
Let m be an odd number V(1) o
+ V(2) o
and
+ V(3) o
+ V(4) o
={_+4
Then: I. F o r
2
"
Xi/2
an a r b i t r a r y
=
_ (i)
4 4 3. Z X 2 = I i= I i i=I
COROLLARY
I+2
(m-!)/2 Z ~=1
and
(2.12) Note
chine
that
3
for Y =
I
(2.32)
(2.34)
= ~ qi
'
i=
(2.35)
that
owing
. Then
solutions
of
system
1,2:3,4
(2.35)
system
for the
(2.12)[120].It
following
47 W i l l i a m s o n Baumert, 27,
29
to p r o p e r t i e s
orders
Golomb,
Baumert
Yamada[145].
(p+I)/2,
7. m = p ( p + 1 ) / 2 , Let us define Baumert,Golomb result
p~1(mod
4)
p~1(mod
have
in H a d a m a r d
92 = 7 2 + 5 2 + 3 2 + 3 2
of
does
not
investigations
solution
of
m:
found
of
all
solutions
[28].
is the o r d e r
of a p r i m e
(Turyn,
is t h e o r d e r
of a p r i m e
(Whiteman,
emphasized
matrices
92 = 9 2 + 3 2 + 1 2 + I 2 r e s u l t s
Further
4)
has
b y L I a set of o r d e r s
and Hall
the
[28].
5. m = 29,37, 6. m =
is k n o w n
for m a -
H a l l [29].
Baumert
41
ti, k is e a s i e r
[120].
4. m = 3 , 5 , . . . , 2 3
sumption
i=I,2,3,4
'
2 2 2 2 4m = q1+q2+q3+q4
Let
system
(2.35)
3. m = 25,
on
zero.
are e q u a v a l e n t .
2. m = 23
on
is n o t
v(i)
j=1
t i , k (Yk+Ym-k)
I. m = 37,
(2.10)
of ~
2 ~i = 4 m
2.1.
processing
equation
(m-l)/2 ~
+ 2
v°
k only one element
m from that
not
[120]. F o r
in H a d a m a r d
items
1-7.Note
also
1971) that
all of p r e s e n t a t i o n s
example,
matrix
1972).
the p r e s e n t a t i -
whereas
the p r e s e n t a t i -
[120]. system
(2.10)
were
carried
out
o n the
as-
20
4m :
So,
it w a s
result type
out
proved
in
in H a d a m a r d has
Generalization
of
two
4m
= x
of
m=29,
12
+ y
2
+ x
37,
+ x2 + y2
2
+ y
2
the
order
+ y
2
+ y
2
+ y
2 2
presentation 104
of
whereas
the
first
type
does
presentation
of
not thi~
41.
Williamson
of
conditions
- alteration
of
number
said
2.1
t,q e o r e m
to
be
AA T
(2.1),
of
[ 295].
has
been
generally
carried
(2.2}.
matrices.
Square
Williamson
I. M N T = N M T 2.
2
that
- alteration
m are
+
directions:
DEFINITION
and
= x
matrix for
in
4m
[145]
solution
12
[-I,+I)
matrices
matrices
A,
B,
C,
D of
order
provided
M,N6{A,B,C,D}
+ BB T
+ CC T ~
[2.36)
DD T = 4mI
(2.371 m
Note
that
with
conditions
automatically In can B,
1974
be C,
those
and
D and of
cnndition
7.Wallis
satisfied has
(2.11
[ 288]
both
noted
such
matrices
that
7
condition
12.36)
holds
12.3) . conditions
and
matrices
(item
the
becomes
non-circulant
constructed
Wi]!iamson
(2.2)
(2.37) has
for
and
of
) with
(2.36)
non-syn~etric orders have
and
matrices
coinciding been
(2.37~ A.
with
constructed
by
9~iteman. At
present
the
Wil]iamson
matrices
of
following
orders
have
been
constructed: 1. m ~
100
2.
9k
3.
m(4m+3)
4.
93
5.
2m,
6.
(m+11 (m+2),
symmetric
k
with
exception
is a n a t u r a l
number
. m(4m-1),
mC{1,3,5
(Wa]lis m
the
is
[311 ] .... , 2 3 , 2 5 }
(Wallis
1975) .
[311 ]~
the
Hadamard
35,39,47,53,67,71,73,83,89,941295]
order
of
existing
m~1(mod
4)
is
matrix
[295 ].
Wil]iamson
a prime
and
m+3
matrices is
the
(Wallis[ order
of
311]) some
21
7. 2.39,
2.203,
6a I
8.
10 a 2
a. > 0, a r e
2.303,
- 14 a 3
•
non-negative
2.333,
2.669,
18 a 4
22a5
from where
2.1603
• 26 a 6
(Wa]lis
. m,
in p a r t i c u l a r
[295]).
mEL 1 ,
i=1,2,.
Williamson
.,6,
matrices
of
l-
order
2.35,
2.65,
9. m k ( m + 1 ) ,
2.77
are
m~1(mod
obtained
4)
is the
(Sarukhanian,
order
1978)
of a p r i m e
number,
k~0
Spence,
m satisfying
the
items
1977). 10.
3k
7.3 k, k>0
11. L e t u s d e f i n e
(Mucho~adhyay
[327])
b y L 2 a set of n u m b e r s
I-I0. 1
12. m ~i()2,n
, where
m,nEL,~ i are
arbitrary
non-negative
integers
1
L=LIUL 2 In
(Agaian,
Sarukhanian
1965 C o e t h a l s
trictions
(2.0)
(Such m a t r i c e s
and
and
Seidel
(2 37}
have
been
called
in c o n s t r u c t i n g
matrices
with
such properties
(a,b,c
of n o n - c o m m u t a b i l i t y
hals-Seidel analogu~
array
of T h e o r e m 2.4
THEOREM
del matrices der
instead 2.1.
have
m.
the
later
conditions
of
with
(2.1),
res-
(2.2).
ones.)
They
succe-
m, m E { 3 , 5 , . . . , 6 1 , 2 a . 1 0 b . 2 6 c + 1 }
are n o n - n e g a t i v e of
the m a t r i c e s
Goethals-Seidel
of o r d e r
such matrices
integers
authors
t h a t of W i l l i a m s o n
[111-113]).
have
Be-
to u s e G o e t -
for ~ r e s e r v a t i o n
the
It h o l d s
(Goethals-Seidel
of o r d e r
discussed
discarding
eded
cause
[~I]) .
Then
[111]).
array
GZ
Let A,B,C,D
[4]
be Goethals-Sei-
is an H a d a m a r d
matrix
of o r -
4m. In
[297]
Theorem taken
2.4.
Wallis So,
and Whiteman
matrices
back-circulant,
An Wallis (number
A,
instead
generalisation
in
Instead
of c o n s t r u c t e d matrices)
Williamson
and Goethals-Seidel
generalized
and matrix
[4] t h e y
discussed
BY[4].
Williamson
that
as
used large
array
is a r r a y
were
was proposed
were
times
of W i l l i a m s o n
matrices
method
matrices
is t h r e e
ones,
modifications
circulant
of W i l l i a m s o n
instead
other
taken
of W i l l i a m s o n
Williamson
called
of GZ
F-matrices
and
obtained
B, D w e r e
important [299].
have
WA
analyzed
C was
by
F-mafrices as t h a t
synthesized [4]. in
of
[6,
of from
Finally, 167,
so
208]
22
(The g e n e r a l i z a t i o n replaced ralized
Williamson where
m6L,
number
From logues
of
analysis
where
analoques
and construct - find Williamson
m6L1,
matrices
Now we
turn our that
in
DEFINITION
we c o m e
are matural
which
formulae permit
numbers.
find
matrices
matrix
give
with
2.1.
those -
gene-
that n the
and theorems,
and ana-
questions:
a notation
containing
investigate
to c o n s t r u c t
to the
modifications
2.2.
of c o n s t r u c t i n g
A set of
solution
of
a notation
(-1,+I)
(0,+I)
matrices
matrices matrix
infinite
classes
decomposition
of W i l l i a m s o n
of Williamson
is a s q u a r e
i.e.
of n e w g e n e r a l i z e d
questions
stated
of W i l l i a m s o n
s W W~ ill
The
notation
above.
families
con-
matrices. {W i} i=II
of o r d e r
(s 1 , s 2 , . . . , s l , B m , m )
B m of o r d e r
m will
provided
m, B m ~ 0 s u c h t h a t
(2.38)
1 = M X s I i= I i m
of
(2.39)
family
of w i l l i a m s o n
matrices
of matrices
of
holds
1 X i=I
Williamson
of H a d a -
into product
w.swT w.swT i m 3 3 m I
NOTE
Note
for a g i v e n
to s t u d y of f o l l o w i n g
factorisation,
i,j=1,2,...l,i@j
2.
The
known.
of W i l l i a m s o n
matrices
[5 ] w a s p r o p o s e d
a family
for e v e r y
ar~
is a n a l y z e d :
Williamson
attention
all known
I. T h e r e
matrix-blocks).
are
matrices.
of W i l l i a m s o n
allowing
multipliers.
be called
a,b,c
problem
theorem
such recurrence
sparse
taining
orders
matrices
them.
mard matrices
Note
following
of a l l m o d i f i c a t i o n s
- for n e w g e n e r a l i z e d all known
from circulant
of W i l l i a m s o n
of Williamson
symmetric
n 6 {3,5,...,59,61}
in f a c t a f o l l o w i n g of all k i n d s
that circulant
ones
matrices
(2a10b26c+1)m,
in[145]
here
by block-circulant
-mn,
-
means
for
1=4,
s1=s2=s3=s4=1,
Bm=I m
coincides
2 3
-
8 Williamson matrices
for 1=8, s1=s2=s3=...=s8=1,
-
Yang matrices
-
Williamson matrices obtained by Turyn,
-
Goethals-Seidel
-
Generalized Williamson matrices
for 1=2, s1=s2=1,
matrices
Bm=I m
BmI m
for 1=4, s1=s2=s3=s4=1,
Bm=R m
for 1=4, s1=s2=s3=s4=1,WiWj=WjWi •
Following theorem is true THEOREM 2•5 " Let {W~ i=I 1
be a Williamson
m) and there is an orthogonal consisting of elements
design of type
family
(Sl,S2,..•,Sl,I m,
(Sl,S2,...,s I) of order n
~xi, xi~0. Then there exists an Hadamard matrix
of order mn. NOTE 2.2. All principal ces in particular, Yang
theorems
Williamson
for constructing
of Hadamard matri-
(1944), Baumert-Hall(1965),
(1971), Goethals-Seidel(1967)
Wallis
(1976),
theorems are special cases of theo-
rem 2.5. We note some properties PROPERTY 2.1. Let
of family of Williamson matrices•
(W I, W2, W3, W 4, Bm, m) be a Williamson
family•
Then a)
(11WI, 12W2, 13W3, 14W 4 , Bm,m)
is a Williamson
family,
b)
(WIXH, W2xH, W3xH , W4xH, BmXIn, mn)is a Williamson
i i = +_I
family,
if H
is an Hadamard matrix of order n. PROPERTY 2.2. Let
(WI, W2, W3, W4, Bm, m) be a Williamson
family•
Then a)
(W liT, w2JT, ~3 T, w41T, B m,m) , where i,j,k,l=0,1,
b) there exists a Williamson PROPERTY 2.3. Let
family
W °Tp = Wp, wIT=w T p P
I W3, I W4, I I2m, 2m). (W I , W2,
(WI, W2, W3, W4, Im, m) be a Williamson
Then there exists a Williamson
family
family.
1 W3, I W4, I I2m, 2m). (W I , W2,
Now let us introduce a theorem about existence of Williamson families special cases of which were proved in [44]. n-1 THEOREM 2.6 • Let {Wi}4i=I ' Wi = Z Ai, ~ U k , n is an odd number, k=0 Ai, k are square
(-I,+I) matrices of order m. Then for
(WI, W 2, W 3, W 4,
24
Bm×Rn,
mn)
be a W i l l i a m s o n
n-1 X k=0
Z Ai,kA~,k)l i= 1
n-1
4
4
X
X
k=0 where
family,
i=I
it is n e c e s s a r y
and sufficient
that
co
= 4mnl
(2.40) mn
T A. ,A, +, , I,K ±,n K-3-1 (mod n)
=
0
(2.41)
j=0,1,2,...,(n-1)/2.
NOTE sary a n d (2.0),
2.3.
sufficient
(2.1)
NOTE
For A i , k 6
and
{-I,+I}
conditions
for e x i s t e n c e
(2.40)
of W i l l i a m s o n
and
(2.41)are
matrices
neces-
satisfying
(2.2)
2.4. M a t r i c e s W I = jxI+AIX(U-U2_U3_U4U5_U6 ) , W 2 = AIX(I+U-U2+U3+U4-U5+U6 ) , (2.42) W 3 = J x I + A 2 x ( U - U 2 - U 3 - U 4 - u S - u 6)
,
W 4 = A2x(I+U-U2+U3+U4-U5+U6 ) where
f i r s t rows of c i r c u l a n t (-t
1 -1
(-1
-t
satisfy conditions
1
1 -1
1 -1 -1
(2.40)
1
and
matrices -1 1
-1 1
(2.41)
A I and A 2 are r e s p e c t i v e l y
-1
1
1 -1
1 -1 -1
1)
1 -1)
a n d are W i l l i a m s o n
matrices
of or-
der 91. THEOREM n)
11,
family
2 • 7. L e t
(A I , A 2
,-.-,All
12 = 2,4 be W i l l i a m s o n (WI, W 2 , . . . , W k ,
PROOF. CASE
families.
Imn , mn),
We w i l l c o n s i d e r
, Im, m) and
(B I
,
B2,
.-.,B12
Then there exists
,In,
a Williamson
k=2,4,8.
3 cases.
I. 11 = 12 = 4, k = 8
Introduce
operators
V I = V 2 ( X I , X 2 , Y I , Y 2) = [ X I X ( Y I + Y 2 ) - X 2 x ( Y I - Y 2 ) ] /2 V 2 = V 2 ( X I , X 2 , Y I , Y 2) = [ X I X ( Y I - Y 2 ) + X 2 x ( Y I - Y 2 ) ] /2
(2.43)
25 Put W i : Vi(AI,A2,BI,B2)
, Wi+ 2 = Vi(A3,A4,BI,B 2)
Wi+ 4 = Vi(AI,A2,B3,B 4) Let us show that is introduced
(WI, W2,...,W8,
matrices
Let us examine
satisfy
T I T VIV I = ~[XlXlX(YI+Y2)
, Wi+ 6 : Vi(A3,A4,B3,B4),
Imn , mn)
(2.38)
and
i=1,2
family,
that
(2.39).
(2.39).
(YI+Y2) T+x2x~x (YI-Y2) (YI-Y2)T-2xIX~(YIYI T - Y2Y~)]
1 T V2V ~ = ~[XIXIX(YI-Y2)
is a Williamson
the conditions
the conditions
,
(YI-Y2)
T+
,
T X2X2x(YI+Y2)
T T (YI+Y2)T+2xIX2×(YIYI -
- Y2Y~)]. NOW,
since (YI+Y2) (YI+Y2)T
+ (YI-Y2) (YI-Y2)T
=
T
T
2(YIYI+Y2Y2 )
then T+ T I T (yiy~) VlV 1V2V2= ~[XlXl x I
T+
T T ÷ X2X2 x (YIYI+Y2Y2) ]:
T
2 (XlXl X2X2)x(YIYI
T+
T
(2.44)
Y2Y2 )
Hence, 4
1
4 AiA~x(BIB TI + B2B~ )
1=1 4
T _ I 4 E Wi+4Wi+ 4 2 Z i=I i=I
SO, 8
Z WiW i=I
4
4
'=
) x ( E BiB i=I
= g (
)
26
Further,
since A i and Bi,
i=1,2,3,4,
form Williamson
families,
then
8 I w.wT = 8toni i=lll mn N o w let us examine
the condition WiW ~ = WjW~,
itj=l,2 ..... 8. Note
that T T T T T T +XiXT×(YiY1+2YiY2+Y2Y2) VlV °¼[XlXiXIYiYi-Y2Y21
_x2xT×~ (YIY1_2Y I T Y2T +
T T +Y2Y)x2xxIYiYi-Y2Y21] _ IT T_ y 2 y ~ ) _ X i X 2I× ( Y i Y 1T_ 2TY i Y 2T+ Y 2 Y 2T) + X 2 X 1 ×T ( y V2V TI = ~[XlXl×(YIY1 +
Hence,
y~+
T+ T T T T 2YIY 2 Y 2 Y 2 ) - X 2 X 2 × ( Y I Y I - Y 2 Y 2 ) ]
from definition
Wi,
i=1,2,...,8
and conditions
we have W WT = W wT , i,j=1,2 ..... 8. i 3 3 i The theorem
is proved for 11 = 12 = 4, k=8. Cases 11,12=2,4,
can be proved on the analogy. COROLLARY
2.2•
symmetric matrices are 8-symmetric COROLLARY
This completes
of order m and n respectively,
Williamson
matrices
2.3. There exist
,
(Wallis,
Williamson matrices
and 2-elementa!
8-Williamson
matrices of orders
be a symmetric
(Wallis
[273]).
[219]).
there exist F-matrices symmetric hyperframe
family
Let Ai×Bi,
of order
i=0,I,2,...
T H E O R E M 2.8. Suppose
exists a Williamson
i=I,2,...,8,
(see note 2.1) •
q~l(mod 4), p~1(mod 4) are prime numbers
11 • 7 i , i=1,2,...
PROOF.
then Wi,
are
P~1(mod 4)is a prime power.
-(2r)imk where r,m,k6L,
tion 9)
of order mn
8-symmetric
Note that there were c o n s t r u c t e d
-7 i+I
the proof of the theorem.
If we suppose besides that A i and Bi, i=I,2,3,4,
(p+1)mn, w h e r e m,n6L,
-q(p+1)/2,
k=2,4
of order k. Then there
(WI,W2,W3,W4,Rm×InXIk , mnk).
i=1,2,3,4
hyperframe
of order ran (see Defini-
are F-matrices
of order mn. Let {X,Y}
of order k. Consider matrices
27
Prove Let
that
us
W I = AIXBIXX
+ A2xB2xY
,
W 2 = A2xB2×X
- AlXBlXY
,
W 3 = A3xB3xX
+ A4xB4xY
,
W 4 = A4xB4xX
- A3xB3xY
matrices
Wi,
i=I,2,3,4
form
the
desired
Williamson
family.
calculate
2 WIW2=AIA2xBIB2XXI-A
2 2 2 2 I ×B1 ×XY+A2xB2 xYX-A2A1 xB2B 1 ×X2
W 2 W I = A 2 A I x B 2 B I xX2+A22 xB2 2 xXY-A21 xB21 ×YX-A1A 2 x B I B 2 xy2 From
comparison
prove
the
relations
we
have
one
that W.W. = W W i 3 31
are
WIW 2 = W2W I . By analogy
, i,j
= 1,2,3,4.
hold. Now
let us calculate T
T
T
2
2
T
T
T
2
T
T
T
T
the
relations
Wl ( R m x I n x I k )W2 = A1RmA2XB1B2xX -A1RmAlxB1B lxxY+A2RmA2xB2B2 × T
T
x Y X - A 2 R m A I × B 2 B I xy T
T
T
2
,
W2(RmXIn×Ik)W1 = A2RmAlXB2BlXX +A2RmA2XB2B2xXY-A1RmAlXB1B1×YxT
T
that
all
- A I R m A 2 ×B 1B2 ×Y Similarly
one
can
prove
Wi(Rm×InXIk)W
T
T
T
2
T = Wj(RmXinxIk)W
2
T
T
T
, i,j=1,2,3,4
T
T
T
T y2
W1W1 = A1AlXB1BlXX +A1A2×B1B2xXY+A2AlXB2BlXYX+A2A2×B2B2x W2 W T2
are
T T 2 T T T T T T 2 = A2A2xB2B2xX -A2A lxB2B lxxY-A1A2xB1B2xYx+A1AlxB1BI×Y
true. From
summation
of
obtained
T
T =
WtWI+W2W2 By
analogous
calculations
relations T
T
we
find
T
T
(A1AlxB1BI+A2A2xB2B2 ) x (X2+y 2) we
get
can
28
T = W3W ~ + W4W 4 By
summation
both
sides
4
T T T T (A3A3xB3B3 + A 4 A 4 x B 4 B 4 ) x ( X 2 + y 2 )
of o b t a i n e d
equations
we h a v e
4
E W wT = E (AixBi) ( A i x B i ) T x ( x 2 + y 2) i=I I I i=I The
theorem
is p r o v e d .
COROLLARY
2.4.
(p+1)m,
where
a prime
power.
There
son m a t r i c e s
consider
f r o m the
DEFINITION
2.3.
of f o r m
I. T h e r e element
3 that
Williamson
F-matrices
matrices
of o r d e r
and p~1 (mod 4)
is c o n s t r u c t i n g
is
the W i l l i a m -
ones. square
k=1,2,...,n
exists
in e a c h
1 E w wT i=I i i
2.
NOTE
2.5.
family
Williamson For
a
row
will
(0,-I,+I)
(column)
pendent
ml -n
=
For
{Wi }Ii=I m a t r i c e s be
called
of o r d e r
a parametric
with
ele-
Williamson
Bm,
Bm~0,
occurs
only
then
that
(2.45)
then
give
introduce
matrices
Wi,
the n o t a t i o n
Yang
of p a r a m e t r i c
coincides
with
Willi-
that
of
Williamson
notations: 1=4 w i l l
be c a l l e d
matrices;
for
1=2
de-
they
matrices. W W.:W W , i,j=1,2,...l, i ] 3 1
generalized
two e x a m p l e s
matrices.
following i=I,2,...,i,
parametric
matrices
on p a r a m e t e r s
N o w we w i l l
such
, i,j=1,2 ..... 1
i=1,2,...,n
let us
parametric
If Bm=Rm,
once,
m non-zero
nE X~ I i=I 1 m
on n p a r a m e t e r s ,
be c a l l e d
of o r d e r
(A1A2,...,A1,Bm,m) .
family
ric W i l l i a m s o n
of w h i c h
= W.B wT ]ml
X.=+I, i -
simplicity
dependent
matrix
(Wi,W2,...,Wi,Bm,m,tl,tl,...,~1)
I. If Bm=I m,
2.
question
same
W.B wT im3
will
of e x i s t i n g
A set of
~Xk,
generalized
( W ] , W 2 , . . . , W 1 , B m , m , x l , x 2 , . . . x n) p r o v i d e d
family
amson
exist
m is the p r d e r
N o w we w i l l
ments
= 4mnkImn k
parametric
of d e p e n d e n t
will
Williamson
be c a l l e d
matrices.
on 4 p a r a m e t e r s
paramet-
29
EXAMPLE
2.2.
Q (a,b)
Then
Let
=
a
b
b
-b
b
b
a
-b
-b
-b
-b -b
b
-b
a
b
b
-b
-b
-b
b
a
-b
-b
b
-b
b
-b
a
b
-b
-b
-b
-b
b
a
(2.46)
matrices
A I = Q(Xl,X2)
, A 2 : Q(x2,-Xl)
, A 3 = Q(x3,x4)
,
A 4 = Q(x4,-x3)
are
parametric Note
Williamson
that
dependent
matrices
on
EXAMPLE
matrices
A I and
2 parameters 2.3.
A I (a,b,c)
of
dependent
A 2 are order
on
4 parameters.
parametric
Yanq
matrices
6.
Matrices
=
cJ
c
a
b
b
c
a
A2(a,b,d
b
-a
d
b
-a
-a
d
b
) =
d I
I (2.47)
A3 (a,c,d)
=
c
-
A 4 (b,c,d)
d
c
-b 1
-b
d
c
c
-b
d
-a
are
dependent
ces
of
order
EXAMPLE BixJ 3 are of
order
of
Matrices
parametric 3 i+I
let
parametric
that
4 parameters
square
generalized
parametric
Williamson
matri-
3. 2.4.
Ao Now
on
I
Williamson
matrices
dependent
on
, Ai+1
=
2 parameters
with
= bJ 3 us
Bi+l=Ci+1=Di+1=Aixi3-BixU3+BixU~
, Bo
= Co
= Do
consider
existence
Williamson
matrices
(0,-I,+I)
matrices
= aI 3 - b U 3 + b U ~ (necessary dependent
Vi,j,
and on
sufficient
conditions)
4 parameters.
i,j=1,2,3,4,
of
order
Suppose m
satisfy
30
following conditions: I . Vi, k * V 3 ,P = 0 ' k~p, k,j,p=1,2,3,4. 2•
4 ~ k=1
3.
4 T = mIm' ~ Vk iVk,i k=1 '
4.
4 Z Vk, i , k=I,2,3,4, i=I
(Vk,i VTk,j
vT
+
Vk,j k,
i) = 0
j=1,2
3.
'
i=1,2,3,4.
is
(-I,+I)
5. Vk, l.BmVTp,l' = Vp,iBmVT,i' T 6. Vk, iB mVp,j
i=j+1,
'
+ Vk,jBmVp,i
matrix
(2.48)
i,k,p=1,2,3,4
= Up,iBmVT,j
+ Up,jBmVk, i , i~j,
i,j,k,p=1,2,3,4
7. Vp,iVk, i = Vk,iVp, i , i,k,p=
1,2,3,4
8. Vp,iVk, j + Vp,jVk, i = Vk,iVp, j + Vk,jVp, i, i,j,k,p=
(Wi,W2,W3,W4,Bm,m,al,a2,a3,a4)
it is necessary and sufficient
Vi, j , i,j=1,2,3,4
I-6
for B
b) items
I-8
for B
m m
= I = R
(0,-I,+I)
matrices
m the first part of this theorem.
i= 1,2,3,4 be parametric
order m. Write them in following
Prove that matrices Vk, i satisfy of items
W i l l i a m s o n matrices
of
form:
Wk = Wk(a1'a2'a3'a4)
liamson matrices.
of
m
At first we will prove
Let W1(al,a2,a3,a4),
that validity
existence
Williamson
of order m satisfying
a) items
NECESSITY.
i~j,
1,2,3,4.
T H E O R E M 2.9. For existence family
p~k
4 = i=~laiVk,i
items
1,4 follows
I-6 from
(2.49)
(2.48). Note at once
from definition
Now let us verify validity
of parametric
Wil-
of items 2,3,5 and 6. Cal-
31
culate 4 2 VT 3 4 -i~laiVk,i p i + Z ' j=li=j+1
WkW
!la~.
=
WpW
vT Vp,i k,i
+
i
But WpW k = WkW ~
(2.5o)
a i a j ( V k , .iV T T i) P,3. +Vk,jVp,
X3 £4 3=I i=j+1
ajai(Vk,3
for every k,p=1,2,3,4
.VT VT p,i + Vk,i p,j)
(2.51)
and for every ai,a j hence,
VT .VT Vp,i k,i = Vk,l p,i VT VT Vp,i k,j + Vp,j k,i
= vk,iv , j + vk,jv , i , i j, (2.52)
i,j = 1,2,3,4
Further,
using B m = Im and supposing p~k we get validity
and 6. It is easy to note that for p = k and 3. Indeed,
from
(2.51)
for p = k
4 a2 vT 3 W WT = ~ Vp + ~ P P i=I i ,i p,i j=l from where according to condition
of items 5
we have validity
of items 2
we have
4 E ajai(Vp, .VT VT i=j+1 3 p,i + Vp,i p,j)
4
wT
4
I W = m E a2I p=1 p p i=I i m
we conclude:
4 [ V VT = mI , i = 1,2,3,4 p=1 p,i p,l m
4 T + V p,i Vp,j) T (Vp,j V p,i
= 0
p=1 where
i
=
i+I,
j= 1,2,3.
SO, first part of the theorem is proved.
Second part and sufficien-
cy can be proved by analogy. COROLLARY
2.5.
exist g e n e r a l i z e d ces of order k.
If there exist T-matrices dependent
on 4 parameters
of order k, then there parametric
Williamson
matri-
32
Indeed,
let T I , T 2 , T 3 , T 4 be T - m a t r i c e s
of o r d e r k.
Introduce
follo-
wing notations
Vl, I = T I , V I , 2 = T 2 , V l , 3 = T 3 , V I , 4 = T 4
,
V2, I = T 2 , V2, 2 = -T 2 , V2, 3 = T 4, V2, 4 = -T 3 , V3, 1 = T 3 , V3, 2 = -T 4 , V3, 3 =-T I ,V3, 4 = T 2 , V4, 1 = T 4 , V4, 2
One can v e r i f y Hence,
that matrices
according
are generalized THEOREM
T 3 , V4, 3 = -T 2 ,V4, 4 =-T I
Vi,j,
to t h e o r e m parametric
2.10.
2.9 m a t r i c e s Williamson
If there e x i s t s
m , a l , a 2 , a 3 , a 4) a n d a s y m m e t r i c c2,c3,c4),
then exists
i,j=I,2,3,4 Wk
a Williamson
a Williamson
from
I-8(2.48). (2.49)
of o r d e r k. family
family
family
items
, k=I,2,3,4,
matrices
Williamson
satisfy
( A I , A 2 , A 3 , A 4 , B m,
(CI,C2,C3,C4,In,n,cl,
( W I , W 2 , W 3 , W 4 , B m X I n , mn,c I,
c2,c3,c 4 ) PROOF.
From theorem
Pn.i
satisfying
vely.
Consider
first
6 conditions
following
Qj,k One can p r o v e Hence,
2.9 t h e r e e x i s t
matrices
and all c o n d i t i o n s
Vk, i a n d
(2.48)
respecti-
matrices.
4 = E V9 i=I Pi,k ,i '
that matrices
from theorem
(0,-I,+I)
Qj,k
k j = 1 2,3,4 ' '
(2.53)
s a t i s f y all c o n d i t i o n s
(2.48).
2.9 m a t r i c e s
4
W 3 = k=1 E
f o r m the W i l l i a m s o n theorem
family
(W1,W2,W3,W4,Bm×In,mn,cl,c2,c3,c4)
. The
is p r o v e d .
F r o m note
2.5 a n d t h e o r e m
COROLLARY
2.6.
2.10
If t h e r e e x i s t s
I m , m , a l , a 2 , a 3 , a 4) a n d family
CkQ j ,k ' j : ],2,3,4
follows: Williamson
(Cl,C2,C3,C4,In,n),
(WI,W2,W3,W4,I
i,nml) , i = I , 2 , . . . nm
families
( A I , A 2 , A 3 , A 4,
then t h e r e e x i s t s a W i l l i a m s o n
33
It is known order
[320] that there exist Wi!liamson
matrices A,B,B,B of
7.
Now from example COROLLARY
2.4 and theorem
2.10 follows
2.7. There exist Williamson
type matrices
of order
7"3 l,
i=0,I,2,... THEOREM 2.11. In, n,a,b,c,d)
If there exists a Williamson
and a 2-elemental
exists a parametric
Williamson
hyperframe
family
family
(Ao, Bo, Co, Do,
of order k, then there
(Ai,Bi,C i,Di,Inki,
nkl,a,b,c,d) ,
i=0,I,2,... PROOF.
Let X,Y be a hyperframe
of order k. Consider matrices
A i = Ai_ixX
+ Bi_I×Y
,
B i = Bi_lXX - Ai_ixY
C i = Ci_ixX
+ Di_ixY
,
D i = Di_ixX
Williamson
COROLLARY nH(2ni)a ~ _
matrices
on 4 parameters
of order nk l, i=I,2,...
2.8. There exist Williamson
where n,ni6L
(2.54)
- Ci_ixY
One can show that matrices Ai, Bi, Ci, D i are dependent parametric
,
type matrices
of order
(set of numbers not satisfying conditions
of
i
items
1:10),
aiis a non-negative
COROLLARY
2.9. There exist Williamson
ders 2n, where n 6 V, V ={35, 87, 93, 95, 99, 105, 145,
147,
209, 215,
155,
integer.
161,
217, 221,
111, 165,
even number or-
37, 39, 43, 48, 51, 55, 63, 77, 81, 85,
115, 169,
type matrices
117, 171
119, 175,
121, 185,
225, 231, 243
247, 253,
125,
129,
133,
187,
189,
255
259, 261,
135,
195, 203
143, 207,
273
273,
275,
279, 285, 289,
297, 299,
301
315,
319, 323
325,
325, 333
341,
345,
351,
363,
387
391,
399, 403
405, 407, 425
429,
513,
527
529,
357,
361,
377,
437, 441, 455,
459, 473,
475, 481
483, 493,
551, 555,
559,
561,
567,
575,
583
609, 621, 625
627, 637, 645
651,
667,
675,
693,
713,
725,
729,
759
775,
777,
783
817, 819,
837,
851,
891, 899,
903,
925,
957,
961
989,
999,
1023,
1089, t147,
1161,
495
1073,
1221, 1247, 1333, 1365, 1419, t547,
2013, 2093, 2275, 2457,2639,
525,
2821, 3003, 3367, 3913}
825
1075,
1081,
1591, 1729,1849,
34
Note 2.303,
that
Williamson
2.333,
2.689,
and matrices in
matrices
2.903,
of o r d e r s
2.915,
2.1603
of o r d e r s
2.35,
2.65,
2.77
if t h e r e
exist
Williamson
2.39,
were
were
2.105,
obtained
obtained
2.171,2.203, by W a l l i s ( 1 9 7 4 )
by
Sarukhanian
[208] . Note
that
re e x i s t 2.9
Williamson
is t h a t
knowing
one
the
2.2.
can
matrices
construct
existence
of o r d e r
Williamson
of W i l l i a m s o n
son m a t r i c e s
theorem
but
and not
Williamson
Goethals-Seidel
matrices
method.
(1944)
m,
of o r d e r
The
then
the-
of c o r o l l a r y n without
2n.
root
of the m a t t e r
(construction)
and
(5), W a l l i s - W h i t e m a n
Value
of o r d e r
investigation
array
of o r d e r
2m too.
matrices
Baumert-Hall-Goethals-Seidel
the W i l l i a m s o n
ons:
type
matrices
its d i f f e r e n t
(6), W a l l i s
is
of W i l l i a m modificati-
(7) a n d o t h e r
ar-
rays. The
idea
of the m e t h o d
(A,B,C,D)
of w h i c h
ter
replacement
their
tain
Hadamard First
They ment,
appears
constructed
of H a d a m a r d
L e t us g i v e Baumert-Hall
direction an a r r a y appearing
matrix
was
row
(column)
Williamson
(J.Wallis
made
but
families
in e a c h
precisely
each such one
element that can
afob-
and Hall
row
3 times.
(column)
That
allowed
(1965). a p-eleconst-
156.
of H a d a m a r d
array
containing
notation
of
(1970)).
2.6.[283 ]. An H a d a m a r d
mxm consisting
by B a u m e r t
containing
of o r d e r
a definition
array
DEFINITION of o r d e r
in e v e r y
by c o r r e s p o n d i n g
in this
p6{~A,~B,~C,~D}
ruction
same
of an a r r a y
matrices.
work
have
is the c o n s t r u c t i o n
array
of the e l e m e n t s
H[m,k,l],
of f o r m
k < m is a m a t r i x
~AI,
~ A 2 , . . . , z A k such
that I. E v e r y ZAI,
row
I elements 2. The
rows
A I , A 2 , . . . , A k are NOTE
2.6.
(column) of f o r m and
of H - m a t r i x
~A2,...,I
the c o l u m n s
elements
An Hadamard
has p r e c i s e l y
elements
of H - m a t r i x
of c o m m u t a t i v e array
becomes
of f o r m are
ring.
i elements
of
form
ZA k.
orthogonal
in pairs,
if
35
a)
a Williamson
b)
a Baumert-Hall
c)
a E - array
E =
array
array
f o r k=8,
The
BX[4t] I=1,
m=4
for k=4,
m=8
l=t,
m=4t
where
X2
X3
X4
X5
X6
x7
X8
-X 2
XI
X4
-x 3
x6
-X 5
-X 8
X7
-X 3
-X 4
X1
X2
X7
X8
-x 5
-X 6
-X 4
X3
-X 2
XI
x8
-X 7
X6
-X 5
-X 5
-X 6
-X 7
-X 8
xI
X2
X3
X4
-X 6
X5
-X 8
X7
-x 2
XI
-x 4
X3
-X 7
X8
X5
-X 6
-X 3
X4
XI
-X 2
-X 8
-X 7
X6
X5
-X 4
-X 3
x2
XI
A k in a r o w
matrix,
I=I,
XI
In a b o v e - m e n t i o n e d ferent
for k=4,
work
J.Wallis
proved
that
(column)
of a r r a y
coincides
(2.55)
if the n u m b e r
with
the o r d e r
of d i f m of the
t h e n m is 2.4 or 8. author
of o r d e r
with
12 w h i c h
Sarukhanian
have
is n o t a n a r r a y
constructed
BX[4t]
an
interesting
and consists
of
array
3 parameters.
iIB~xl,x2,x3)D<x2,-x3,xI) D(x3,x2,-xll D(x3,x1,-x2)I A12
A12(x1'x2'x31 tD x3, x2, x11D Ix3, x2xll BIxI D(-x 3,-x I,-x~D
x31
(-x 3 , x 1 , - x ~ D ( X 2 ''X2 XI'X3)
DIx2, xl,x lI B(XI'X2'X3)
where
B(a,b,c)
Note following supposed gonal x/a
=
that
a
b
c
c
a
b
b
c
a
the array
reasons: that
design
2 2 = q1+q2
of type ' q1'
D(a,b,c)
constructed
firstly,
if m = 4 t ,
,
Geramita,
c
b
c
a
c
a
b
Geramita
numbers,
of
interest
(2.56)
also by
and J.Wallis
then
it is n e c e s s a r y
q2 a r e n a t u r a l
b
is a m a t t e r
t is a n o d d n u m b e r ,
(a,a,x)
=
a
and
(1976)
for e x i s t e n c e sufficient
secondly,
have
of o r t h o -
that
all orthogonal
de-
II
36
sign
of
type
(1972),
(s,s,...,s)
Wallis
(s,s,...,s)
(1970)
of o r d e r
constructed,
theorems m,m=is
the n u m b e r
of p a r a m e t e r s
so m u s t
2,4
be
or 8.
some u n c e r t a i n t i e s author
had
DEFINITION element meters
of w h o s e
the
is Zxi,
A-matrices,
,x~2))HT(x~1)
NOTE
2.6.
,x
Jl)
that
if a n d o n l y
in e a c h
row
on n u m b e r
of A - m a t r i c e s .
will
value
.
2m 1
.
1 X i=I
.
but
aland
m, e a c h on
1 para-
X l , X 2 , . . . , x I holds,
.
,...
(1)x(2)i xi i m
includes
is
theorem(1970)
dependent
of p a r a m e t e r s
.
of A - m a t r i c e s
be c a l l e d
that
even
H ( x l , x 2 , . . . , x I) of o r d e r
i=I,2,...,i
_
only
design
of p a r a m e t e r s
definition
Ix I
orthogonal
is not
of C o o p e r - W a l l i s
matrices
, Plotkin
if 1 = 2 , 4 , 8 ,
in p r o o f
(I) ,...,x I
Notation
suppose
limitations
if for e v e r y
H(xll) ,x~1) ..... x ( 1 ) ) s
...
appearing
Square
to
can e x i s t
appearing
2.4.
permit
To a v o i d
to i n t r o d u c e
Cooper-Wallis(1970)
BX[4t],
(2.57)
Yang
and
Plotkin
arrays. Information can
find
theorem
in p a p e r s about
THEOREM ces
about
hyperframe PROOF.
of a u t h o r
existence
2.12.
of o r d e r
construction and
properties
Sarukhanian
of A - m a t r i c e s
[6,11]. Let us give
one only
m it is n e c e s s a r y
of d e p e n d e n t and
on
sufficient
1 parameters
existence
of
A-matri1-elemental
m.
Let A ( X l , X 2 , . . . , X l )
be
an A - m a t r i x
of o r d e r
m. W r i t e
it in
is a 1 - e l e m e n t a l
frame
form 1 A ( X l , X 2 , . . . , X I). = X x K~ i:I ~ l It is e a s y of o r d e r
m. On the
A ( X l , X 2 , . . . , x I) First
to v e r i f y
that
contrary,
the if
is an A - m a t r i x
valuable
a
of A - m a t r i c e s .
For e x i s t e n c e
of o r d e r
and
contribution
set
{Ki}~= I
{Ki}~= I is a frame of o r d e r
of o r d e r
m,
then
m.
to c o n s t r u c t i o n
of BX[4t]
array
(hen-
37
ce,
of A - m a t r i c e s
too)
troduced
an effective
of order
4t u s i n g
ces
of o r d e r Methods
was made method
by Cooper
and Wallis
of c o n s t r u c t i o n
Goethals-Seidel
array
of t h e
GZ[4]
(1970),
They
Baumert-Hall
and notation
inarrays
of T-matri-
t. of constructing
the T - m a t r i c e s
of f o l l o w i n g
orders
are
known: - m6L 3 :{3,5,7,...,59,61} m = 2a10b26c+1,
-
a,b,c
(Cooper, are
1970)
non-negative
integers
(Turyn,1974)
It h o l d s THEOREM amson Then
matrices
(Baumert-Hall,
of o r d e r
a H(A,B,C,D) Note
were
2.13.
Hence,
taking
following
Let matrices
m and H(a,b,c,d)
matrix
that all known
constructed
1965).
is a n H a d a m a r d BX[4t]
arrays
from T-matrices
into account
A,B,C,D
be a B X [ t ] a r r a y matrix
except
2.12,
we c o m e
to
t.
mt.
(Baumert-Hall,
and Goethals-Seidel
theorem
of o r d e r
of o r d e r
t=3
be W i l l i -
array
1965)
of o r d e r
investigation
4.
of
questions.
- introduce teman
notations
arrays
of n e w a r r a y s
and arrays
of o t h e r
(Goethals-Seidel, orders
4t,
t>1
Wallis,Whi-
and construct
them; construct prove
a theorem
ruction One
of n e w o r d e r s ;
allowing
of H a d a m a r d
excellent
about
survey
on o r t h o g o n a l Hadamard
their
problem.
1979
Goethals-Seidel
array
of o r d e r
has
GZ[4t]
arrays
constructed
theorems
for c o n s t -
(1977).
in our o p i n i o n Sarukhanian 4t t u r n i n g for t=6
of o r t h o g o n a l
and application
and Wallis
because, In
all known
is the n o t a t i o n
construction
of H e d a y a t
designs
to u n i t e
matrices.
of t h e g e n e r a l i z a t i o n s
Information
the
T-matrices
has
Here
one
we w i l l
it w o u l d
find not
notation
array
in
dwell
lead away
introduced
into GZ[4]
can
designs.
at t=1
from of and
i
It h o l d s THEOREM
2.14.
If t h e r e
exist
T-matrices
of o r d e r
t and a Goethals-
38
Seidel array of order 4p, then there exists a BX[4pt] For p=1 theore, formation
2.14 coincides
about construction
with the Cooper-Wallis
of Goethals-Seidel
find in papers of author and Sarukhanian constructed
array. theorem.
arrays GZ[4t]
[1,6,11,210
In-
one can
]. Here were also
Wallis arrays of orders 4.6 k, k=I,2,...
An interesting by Matevosian
ides of c o n s t r u c t i o n
(in print).
the GZ[4t]
array is proposed
He p r o p o s e d also notation of Whiteman
of order 4t and its construction
that leads to construction
array
of new
class of Hadamard matrices. Now we will
introduce
a definition
(BX[4t],GZ[4t],WY[4t],W[4t], DEFINITION
containing
A-matrices,
orthogonal
I. Each element of H ( X l , X 2 , . . . , X l ) - m a t r i x ~X i, ~x Ti, ~XiB k, _+X~Bk, where B k is a (0.-1.+])-matrix one non-zero
2. Each row(column)
Xl,Bt,4t]. hasize
is of form
i=I,2,...,i
of order k, B k # 0 each row
(column)
of
element;
T iXi,
{Wi}~= I
contains precisely
si
' zXiB k , +X~Bk} _
3 • H(XI,X2, .. .,XI)HT (XI,X2,...,XI) T T T
T H E O R E M 2.15. Let
- array provided:
of H ( X i , X 2 , . . . , X l ) - m a t r i x
elements of form P, P6{ZXi,
...,Sl,Bm,m),
designs).
2.5. A square matrix H ( X l , X 2 , . . . , x I) of order 4t will
be called an A [ l , S l , S 2 , . . . , S l , X l , X 2 , . . . , X l , B k , 4 t ]
which contains
all known arrays
1 T = i=isiXiXixI4t Z
be a W i l l i a m s o n
family of type
(Sl,S2,
let us have an array of type A [ l , S l , S 2 , . . . , S l , X 1 , X 2 , . . . ,
Then there exists an Hadamard matrix of order 4mt. We emp-
that this theorem unites all theorems
2.1,
2.2, 2.3, 2.4,
2.5
together. Now let us consider sI,XI,X2,...,XI,Bk,4k]. DEFINITION
the construction
of the array A[l,Sl,S2,... ,
For this purpose
introduce
2.6. A set of
(0,-1,+1)-matrices
be called a family of T-matrices by T(l,n,Bn)
provided:
of type
{Ti}~= I of order n will
(I,TI,...,TI,n,B n) and denoted
39 I. T. , T. = 0 i 3
, i~j,
2 . T I T j = TjTi,
i,j=1,2,.°.,l
3. T h e r e for
exists
i,j=1,2,...,l
a square
i,j=1,2,...,l
(2.58)
(0,-1,+1)-matrix
holds
B
of o r d e r
n
n such
that
T.B T T = T.B T T I n 3 3 n i
1
4.
T. l
i=1
(-1,+1)-matrix (2.59)
5.
1 Z i=I
Note matrices
TiTT
that
= nI n
for
coincides
Denote
lar,
with
that
Ti-matrices
notation
of T ( l , n , B n ) -
of T - m a t r i c e s . of d e p e n d e n t
or a set of all k i n d s
(1,1,1,1),
the
and c i r c u l a n t
b y K a set of all k i n d s
4,8) A - m a t r i c e s (1,1),
1=4
(1,1,1,1,1,1,1,1)
on
1 parameters
of o r t h o g o n a l
. Note
that
designs
K contains
of type in p a r t i c u -
array
a bcdll
and also
(1=2,
Williamson,
b
-a
-d
c
c
d
-a
-b
d
-c
b
-a
Yang,
Wallis
(2.60)
arrays.
It h o l d s THEOREM let t h e r e an a r r a y
exists
Let
an a r r a y
i=1,2,...,i,
Define
Denote
matrices
orthogonal of
this
i=I,2,...,I,
W = M ® T
where
be
T(1,m,Bm)-matrices; Then
there
exists
, mn].
the r o w s Wi,
1=2,4,8
A[I,AI,A2,...,AI,Bn,n].
M be an a r b i t r a r y
the K - s e t .
...,i.
L e t Ti,
A[I,WI,W2,..°,WI,Bmn
PROOF. from
2.16.
design
design
by
of type vectors
(i,I,..°,i) Mi,
i=I,2,
from relation
(2.61)
40
T T [W I, W2,
WT =
From the
equation
scalar
(2.61)
product
of
...,
W
follows
rows
the
~
]
TT
,
that
the
matrix
T [TI,
=
T T2,...,
components
M and
the
of
vector
T
]
vector T,
W are
that
is
W = M T 1
WiWj = WjWi, i , j
Obviously, NOW,
l l
= 1,2,...,1
since T B TT lm 3
T .B T ~ 3ml
=
,
i,j
=
1,2,...,i
SO,
WiBmW ]. : (MiTIBmIT M].I , (MjT) B m ( T T M )T
One
can
see
also
WW T =
that
(M ® T) (M ® T) T =
Now,
replacing
have
that NOTE
= WjB m W T
A i in a r r a y
(M ® T) (M T ® T T)
A[I,AI,A2,...,AI,Bn,n]
A[I,WI,W2,...,WI,Bmn,mn] 2.7.
The
- Cooper-Wallis array,
Hence, particular THEOREM
2.16
theorem
is a n
array
by
matrices
of d e s i r e d
W i, w e
type.
Q.E.D.
contains
(1970)
for
1=4,
m=4,
M
is a W i l l i a m s o n
theorem
(1979)
for
1=4,
m=4t,
M is a W i l l i a m s o n
A is a G Z [ 4 t ] - a r r a y .
the
problem
of c o n s t r u c t i o n
to c o n s t r u c t i o n 2.17.
T(4,m,Bm)-matrices.
Let
the
arrays
{X1,X 2} and {Ao,B!,Co,Do}
Then
Let
us
define
of
type
A comes
in
of T - m a t r i c e s .
there
exist
T(4,mnZ,B
following
matrices:
A i = XI×Ai_ I - X2×Bi_ I ,
be T ( 2 , n , B n) and i)-matrices,
mn
PROOF.
W WT i l
A is a G Z [ 4 ] - a r r a y ;
- Sarukhanian array,
theorem
1 E i=I
=
i=1,2,...
41
B i = XlXBi_ I + X2×Ai_ I , C i = XlXCi_ I - X2xDi_ I , D i = XI×Di_ I + X2xCi_ I One ces
can
of order Note
only
mn
that
matrices, ons
see
induction
that
matrices
A±,
B I, C i, D i a r e
T-matri-
.
there
where some
by i
exists
n £{14,
an
26,
algorithm
30,
38,
0
-I
42,
for
construction
50,
54}.
List
of
the
T(2,nIn)-
constructi-
orders,
I. n = 14 X = V 1 x I 7 + R 2 x ( I 7 + U - U 2 + U 3 + U 4 - U 5 + U 6)
,
Y = I 2 x ( I 7 - U + U 2 + U 3 + U 4 + U 5 - U 6) 2. n = 26 X = V1xI13
+ R2x(I13-U-U2-U3+U4-U5+U6+uT-u8+u9-u10-U11-U
y = I2x(I13-U-U2-U3+U4_U5+U6+U7_U8+U9_UI0_U11_U12
12)
,
)
3. n = 50
X = V2xI25
+ R2x(I25-U-U2-U3-U4-U5+U6-U7+U8-U9-U10-UII+uI2+u
_UI4_uIS+u16+UI7_uI9_u20-U21-U22-U23-U24)
13-
,
y = I2x(I25-U+U2+U3-U4+U5-U6+U7+U8+U9_UI0-UI1_u12_U13_U14-U15+ +UI6+UI7+uI8_UI9+U20_U21+U22+U23_U
STATEMENT exist
Let
Ao,Bo,Co,Do
T(4,k,Bk)-matrices
PROOF. 0 0),
2.1.
(0 -I
Let -I
first 0 -I
with
rows -I)
of
24)
be
k=61m, the
T(4,m,Bm)-matrices.
there
i=I,2,...
matrices
respectively.
Then
XI,X 2 are
Introduce
A i = I6xBi_ I + XlXCi_ I + X2xDi_ I ,
of
matrices
form
(0 0 0 I
42
= I 6 x A i _ I - X I xDi-1
Bi
+ X2Tx Ci-1
'
T C i = I6xDi_ I ÷ XlXAi_ I - X2xBi_ I , Di=I6xCi_l One I
can p r o v e
that
. )-matrices, 61m COROLLARY
{63 65 69
+ XlXBi_ I + X2×Ai_ 1
for e v e r y
i matrices
Ai,
Bi,
Ci,
C i are
T(4,61m,
i=I,2,...
2.10.
There
exist
T-matrices
75 77 81 85 87 91 93 95 99
111
of o r d e r 115
117
2n,
119
n6L2,
123
125
L2 = 129
133
135
141
143
145
147
153
155
161
165
169
171
175
177
185
189
205
209
217
221
225
231
235
243
245
247
255
259
265
273
275 285
287
295
297
299
301
303
305
315
323
325
329
343
345
351
357
361
371
375
377
385
387
399
403
405
413
425
427
429
435
437 441
455
459
465
475
481
483
495
505
507
513
525
533
551
555
559
567
575
585
589
603
611
615
621
625
627
637
645
651
663
665
675
689
693
703
705
707
715
725
729
735
741
765
767
771
775
777
779
783
793
805
817
819
825
837
845
855
861
875
885
891
893
903
915
925
931
945
963
969
975
987
999}.
Following
is a l s o
STATEMENT
2.2.
true
There
a) k=m(p+l) i, w h e r e of e x i s t i n g b)
existing
2.18.
a n d an a r r a y exists
STEP
I. D e f i n e
is a p r i m e
Williamson
p=l(mod
generalized If t h e r e
of type
We w i l l
4)
give
power,
type
m - is an o r d e r
matrices; m is an o r d e r
4)
is a p r i m e
power,
of
exist
only
m - is an o r d e r
T-matrices. a 2-elemental
of type
the
hyperframe
of o r d e r
A[I,AI,...,AI,B4t,4t],
A[l,Wl,W2,...,Wi,B4kt,4kt
a matrix
where
T-matrices;
Ho(A1,A2,...,AI)
an a r r a y
PROOF.
p=1(mod
generalized
of e x i s t i n g
T(4,k,Bk)-matrices
n/26{5,7,13,15,19,21,25,27};
c) k = m ( p + l ) i, w h e r e
THEOREM
exist
generalized
k = m n l, w h e r e
195 203
sketch
then
].
of the p r o o f .
4t P = ( a i , j ) i , •=13
elements
of w h i c h
are
k there
43
a 2 i _ 1 , 2 i = I, a 2 i , 2 i _ 1 = - I ,
for i ~ 2r-I,
ai, j = 0
i=1,2,...,2t
j ~ 2r or i ~ 2r,
j ~ 2r-I,
r=1,2,...,2t
Obviously pT = _p STEP
~ . Denote
, ppT = i4 t by H I a m a t r i x
HI=Ho(AI,...,AI)
(IkXP).
Then
HoH1T + H1Ho T = 0 STEP
~ I . One can p r o v e
statement
of the theorem,
COROLLARY
that
the a r r a y A = XxHo
({X,Y}
is a frame
2.11.
There
exist
Hadamard
2.12.
There
exist
arrays
+ YxH I is that of
of o r d e r
matrices
k) .
of o r d e r
8mnk,
where
m,n,k 6 L . COROLLARY
G Z [ n m i] THEOREM and G o l a y there
2.19.
PROOF.
If there
supplementary
exists
, B X [ n m i]
sequences
Let H ( A o , B o , C o , D o )
length
m. Let us c o n s i d e r
a Goethals-Seidel of length
type a r r a y
m
be G o l a y
following
array
of o r d e r
(see D e f i n i t i o n
of o r d e r
be a G o e t h a l s - S e i d e l
4t and A = {ai}T= I , B ={bi}~= I
types
, W Y [ n m i]
exists
a Goethals-Seidel
of f o l l o w i n g
type a r r a y
supplementary
matrices:
m
=
D 3•
=
I m ~[iE=1
bi) ui_ I (a i
+
m
- Zi=I ( a i
-
m xDj_ I + Z (a i i=I
-
bi) ui-1
ui-1 b i)
xD
3 _I
T xC-I
of o r d e r
sequences
1 m bi) ui_ I m ui-1 T ] Bj = ~[ Z (a i + xBj_ I + Z (a i - b i) xA -I ' i=I i=I
C3•
11),then
4tm I.
I m bi) ui_ I m T Aj = ~[ Z (a i + xAj_ I - ~ (a i - b i ) u i - l x B _i ] , i=I i=I
I 2[iZ=1 (ai + b i ) u i - l x c j - 1
4t
]
]
,
of
44
From
definition
conditions
of
2.3
matrices
definition
2.3.
Aj, Bj, Cj, Dj of o r d e r kmj on
Ao,
Bo,
Further,
Co,
it
also satisfy
Do o f
is e a s y
order
m
to n o t e
satisfy that
the
matrices
t h e c o n d i t i o n s of d e f i n i t i -
2.3. Now
Co,
by
Do
substitution
in a r r a y
H(Ao,
matrices Bo,
Co,
Aj, Bj, Cj, Dj f o r m a t r i c e s Ao, Bo,
Do)
we
can
make
sure
that
matrix
H ( A i,
Bj, Cj, Dj) i s a G o e t h a l s - S e i d e l a r r a y of o r d e r 4tm 3, j = 1 , 2 , . . . The
theorem
is p r o v e d .
Information 2.3. of
about
sequence
one
P a l e y - W a l l i s - W h i t e m a n.......... method.
construction
ley
Golay
skew-symmetric
Hadamard
can
This
find one
matrices
in
of
§ 3,
the
based
I.
first on
methods
following
Pa-
construction:
:nll
H =
IT
(2.62)
n
where mard
A
is a b a s e
matrix
with
such
restrictions
that
about
construction
A structure
an
Hada-
matrix. The
interesting
matrix
(nucleus)
Goethals
and
dification
theorems
belomg
Seidel
of
THEOREM
this
2.20.
to
Ryser
(1967)and method
Let
A,
J.Wallis
belongs B
(1950,
1952, (1970,
to J . W a l l i s
(-I.+I)
matrices
1968), 1971,
AT = A
, BT = B
AB T = BA T
the
matrix
, IN T = 1
, N 6{A,
B}
(1965),
Further
mo-
(1972).
of o r d e r
, A A T + BB T = 2 ( m + 1 ) I
of b a s e
Szekers 1972).
conditions
Then
H remains
- 2J
,
m
satisfying
the
45
I
I
1
1
I
-I
-1
i
A
B
1 T -B
A
H :
(2.63) 1 T -i T 1T
is a
symmetric In
Sekenres
used
set
of o r d e r
m and
of o r d e r
In cial
1970
Hadamard
of
order
2(m+I) .
construction
constructed
(2.63)
the
and
notation
skew-symmetric
of
dif-
Hadamard
mat-
[260]. used
matrices,
2.21.
of
mentary
the
of
the
that
construction is HI,
(2.63)
and
H 2 - Hadamard
constructed
matrices
with
specon-
H2H
THEOREM matrices
4(m+1)
J.Wallis
dition H
tor
matrix
1969
ference rices
Hadamard
type
(Wallis-Whiteman I and
difference
let
set
Z be
a type
4 -{2m+I,
consisting
[311]).
m,
of
Let
X,Y
2 incidence
2(m-I) ]. L e t
+I.
and
matrix
also
length
2m+I
-I
-I
-I
-1
1
1
1
1
I
-I
I
-I
-1
1
-i
1
I
-1
-I
1
-1
l
1
-1
1
I
-I
-1
-1
-i
1
1
iT
1T
iT
iT
A
B
C
D
-1 T
1T
-1 T
1T
-B T
AT
-D
C
-1 T
1T
1T
-1 T
-C
DT
A
-B T
-i T
-i T
iT
iT
-D
B
AT
matrix
of
W be
incidence
of
1 be
supple-
a row-vec-
Then
(2.64)
is a n
Hadamard
Note a base
that
mits
instead the In
8(m+I).
Wallis-~iteman
array
of o r d e r
4 was
taken
here
as
matrix.
In p a p e r but
the
order
-C
mentioned
of W a l l i s - W h i t e m a n
construction
1972
an analog
Wallis
of
the
proposed
of
construction
array
a new
Hadamard a
stronger
array
matrix
of
(2.64) was
taken,
order
construction
was
considered that
4(2n+I)
namely,
per-
2
an
Hada-
46
mard
array
H[4t,t]
the construction Spence
in
(Baumert-Hall
of H a d a m a r d
array)
matrix
as a b a s e
of o r d e r
1975 u s e d G o e t h a l s - S e i d e l
array8
that
allows
8. (m+1) .
array
of o r d e r
4 as a b a s e
array. So
in a l l c o n s t r u c t i o n
array
(for e x a m p l e ,
Wallis-Whiteman Hall
in
array
(2.63)
ting
and
supplement
in a n H a d a m a r d
4,
arrays the
essentiality
it is k h e
of o r d e r
and Goethals-Seidel
perties
mentioned
in l a s t
of o r d e r
some number
matrix.
Yang
This
is u s a g e
array,
in
two cases of order
of a base
(2.64)
there
4 with
it is t h e
are Baumertdefined
of n e w r o w s a n d c o l u m n s
is t h e
pro-
resul-
i d e a of P a l e y - W a l l i s - W h i t e m a n
method. Note
that
- in c o n s t r u c t i o n s Williamson Hadamard mal
-
family,
ml,
where
N6{A,
Some
of
different
find
Spence
eldy
theorem. very
a GZ[4]-array THEOREM order t-1
v,
times
and
result
the a r r a y
[64]
containing condition
become the maxi-
in an H a d a m a r d
ml~ m/2
D}
(2.65)
were
used
not give
was
on the
whole
to construc-
matrices. A,B,C,D the
obtained
Let A,B,C,D
that
in m a t r i c e s
I appears B,C,D
be
and on base
statement
by Whiteman
circulant
in e a c h
it a p p e a r s
AA T + BB T + CC T + DD T = 4(2t
Then
to Cohn theorem
form a
array
of t h i s
(1976),
unwi-
with
array.
[317].
suppose
would
on m a t r i c e s
we will
strong
as a base
2.22.
v=2t
Here
matrices
base
the
Hadamard
restriction
(1975).
Finally,
C,
mentioned
skew-symmetric
can not
matrix
satisfy
B,
A,B,C,D
otherwise
but according
m must
- the constructions tion
the m a t r i c e s
of an H a d a m a r d
of o r d e r
IN T = 1
since
matrices,
order
matrix
mentioned
(-1,+1)-matrices
r o w of A m a t r i x
precisely
+ t)I
- 4J
t times.
of
precisely Let also
47
is a n H a d a m a r d
K
_X T
yT
ZT
WT
X
A
BR
CR
DR
Y
-BR
A
Z
-CR
DTR
W
-DR -cTR
matrix
We e m p h a s i z e
-DTR A
instead
-BTR
BTR
of o r d e r
that
cTR
A
4(2t+I) . of c o n d i t i o n
(2.65)
Whiteman
used
in f a c t
the c o n d i t i o n
1A T = -21
In w o r k m e n t i o n e d rices
A,B,C,D
Recently man
theorem
se a r r a y
with
hals-Seidel LEMMA there
with
only
array
2.2.
exists
same
1B T = iC T = 1D T = 0
Whiteman
of o r d e r Aturian
,
constructed
2t, w h e r e
(to a p p e a r )
t and
restrictions
difference of o r d e r
that
4t,
If A is a n a r r a y
a 4-elemental
and
2.23.
l e t A., 1
i=I,2,3,4
are p r i m e
a theorem
(2.66)
array
mat-
to W h i t e -
A,B,C,D
considered
a n d on b a is a G o e t -
t > I. A[4,t,t,t,t,Ai,A2,A3,A4,B4t,4t], 4 {Ki}i= I such
hyperframe
then
that
4 = ~ K i:I l
L e t A be an a r r a y A = A [ 4 , t , t , t , t , A I A 2 , A 3 , A 4 , B 4 t , 4 t ] be circulant
(-1,+I)
matrices
of o r d e r
fying conditions 4 i=I
(2.66))
numbers.
analogous
on m a t r i c e s
the b a s e
A[4,t,t,t,t,l,1,1,1,B4t,4t]
THEOREM
(with c o n d i t i o n
2t-I
proved
(2.66)
A AT = 4(n+1)I - 4J 1 1 n n
,
where
IQ 2 = IQ 3 = IQ 4 = 0
, Qi 6 { A I , A ~}
, i = 2,3,4
n satis-
48
I Q I = -21
Then
the
,
QI 6 { A 1 , A ~}
.
array
[i 4
4
-K I + ~ K i=2 l
E i=I
K × 1 l (2.67)
4 E i=I
is a n H a d a m a r d Note
that
i=I,2,..,
A[2,
secondly, theorems
matrix firstly,
K. × l
of o r d e r this
can be enlarged
t,
t, AI,
A2,
it c o n s i s t s et al.) .
IT
4t(n+1) .
theorem
with
t o the c a s e
B 4 t , 4t]
all
A
first
modified
when
or A =[8,
theorems
array
conditions A is of
for
A i,
form
t, t , . . . , A I , A 2 , . . . , A 8 , B 4 t , 4 t ]
(Whiteman,
Aturian,
Wallis
49
§ 3. some p r o b l e m s of c o n s t r u c t i o n
for H a d a m a r d m a t r i c e s
In this p a r a g r a p h we will give a survey of general a p p r o a c h e s to the c o n s t r u c t i o n s
for classic H a d a m a r d m a t r i c e s namely, Golay-Turyn,
Plotkin and Wallis approaches. zed and strenthened,
Later these a p p r o a c h e s will be g e n e r a l i -
in p a r t i c u l a r a r o r r e l a t i o n between g e n e r a l i z e d
6-codes and T - s e q u e n c e s will be found, a recurrent ruction of g e n e r a l i z e d
formula for const-
6-codes will be given a l l o w i n g to c o n s t r u c t a
new class of T-matrices,
B a u n e r t - H a l l and Wallis arrays and hence, Hada-
mard matrices.
For example we will prove the e x i s t e n c e of H a d a m a r d matk rices of order 2 S . v I ' V 2 , . . . , V k where V >l 3, s<_ E [21og2(Vi-3) ] -k+1. i=I Second section is denoted to 2 suppositions about Plotkin p a r t i t i o n s D(4m,4)
and D(8m,8) . Here we will prove that Plotkin method does not
always work. 3.1. G o l a y - T u r y n a p p r o a c h and its ~ e n e r a l i z a t i o n s .
This a p p r o a c h is one of poorly hods of c o n s t r u c t i o n s because of p o s s i b i l i t y
i n v e s t i g a t e d but good working met
for H a d a m a r d m a t r i c e s . I t
is interesting not only
to c o n s t r u c t a wide class of B a u m e r t - H a l l and
Wallis arrays but because of p o s s i b i l i t y to c o n s t r u c t H a d a m a r d m a t r i c e s from codes
(before H a d a m a r d m a t r i c e s were used for c o n s t r u c t i o n of co-
des with d i f f e r e n t c h a r a c t e r i s t i c s
for example,
e q u i d i s t a n t codes or
R e a d - M a l l e r codes, but now one can resolve the reverse problem). The a p p r o a c h
is b a s e d on the n o t a t i o n of s u p p l e m e n t a r y
D E F I N I T I O N 3.1.
(Golay,
1960) . Sets {ak }mk:1
sequence.
and {bk} ~=I
consis-
ting of -I and +I e l e m e n t s are s u p p l e m e n t a r y Golay sequences of length m
provided m-j i=]E (aiAi+ j + b B.±1+3') = 0, j = 1,2,..., m-1
(3.0)
50
Information of G o l a y
about
(1960,
in p a r t i c u l a r ,
supplementary
1962),
Golay
Kruskal
sequences
(1961),
Taki
one
can
(1969).
find
These
in p a p e r s
papers
state
that
- m is an e v e n
number
and
is the
j m-j - i=~ I (a.A ~ 1 m + l.- 3• + b i b m + i _ ~)J + i=I
sum of two
sequences;
(aiai+ ~J + b i b i + ~)~ = 0
j = 1,2,...,m-I;
- am_i+ I = qiai -
if a n d o n l y
if b m _ i + I = - q i b i ' qi
= ~1
I [ a i R l ( W 2 i + l ) ]2 + Z [biRl(W2i+1) ] 2 + Z [ l i I m ( W 2 i + 1 ) ] 2 + i6s i6D i6s i[aiim(W2i+1)]2 i6D
m = ~
'
where s = [i-,
- w is the number,
2mth
I m is the
- there length
o<
exist
,
root
of unity,
imaginary
D = {i;
I}
part
constructions
0 ~ i ~ m
R 1 is the real
of a c o m p l e x for
, qi : -I};
part
of a c o m p l e x
number;
supplementary
Golay
sequences
of
m, w h e r e
a) m = 2 a 1 0 b 2 6 c, a , b , c b)
qi
=
i<m
m, m ~ 100 e x c e p t
- there
exists
no
are
non-negative
integers;
m = 58,68,74,82,90,98;
supplementary
Golay
sequence
of
length
different
approach
to u s a g e
2°9a,34,
36,50. Turyn tary
in
Golay
1974 p r o p o s e d
sequences
for H a d a m a r d
L e t V = {Vi}T= I be are
-I or
+1.
is p r e c i s e l y
set of n - o r t h o g o n a l DEFINITION called
3.2.
a n-symbolical
matrices
a set of n o n - z e r o
If m a x i m a l n, n ~ m
some
number
; then V ={Vi}~= I will
supplemen-
construction.
k-vectors
of m u t u a l l y
of
coordinates
orthogonal be c a l l e d
vectors
of w h o s e from V a
(Turyn,
1974)
vectors
w i l l be
vectors. m A set V --{ V i}i=1
8-code
of
of n - o r t h o g o n a l
length
m provided
51
m-j ZiViVi+j i= Leter 6(n,m)
on a n - s y m b o l i c a l
6-code
3.1.
m follows
From existence
existence
existence
by
by 6 ( n , m ) - s e q u e n c e .
of s u p p l e m e n t a r y
of 2 - s y m b o l i c a l
in p a r t i c u l a r ,
(3.1)
of length m will be d e n o t e d
and set V ={Vi }mi=1 will be d e n o t e d
NOTE
have
= 0 , j = 1,2,...,m-I
~-code
Golay
sequence
of length
of length m0 F r o m here we
of 2 - s y m b o l i c a l
~-code
of length
2a'10 b"
•26 c . Note
that
the p r i n c i p a l
6-code one can find some e s s e n t i a l PROPERTY number) exists
then
in a b o v e - m e n t i o n e d
If there e x i s t
of length ml, a 2-symbolical
there
3.2.
paper
about
of Turyn.
a n-symbolical
and a 2 - s y m b o l i c a l
2-symbolical
N o w we will give
6-code
~-code
(n is an even
of length m2,
then there
code of length m l . m 2.
If there e x i s t s
exists
STATEMENT
and t h e o r e m s
ones.
3.1.
PROPERTY
properties
a 4-symbolical
3.1. There
exists
a 2-symbolical 6-code
6-code
of length m,m>2,
of length m/2.
no 3 - s y m b o l i c a !
6-code
of length m,mE-1
(mod 8). Note
also that
every m there Finally,
in this paper
exists
THEOREM
of order
4mt.
From
1975
(constructed)
array
a basic
of H a d a m a r d
of order
(Geramita, m-sequences
(Sl,S2,...,Sl),
sequence
6-code
the
theorem allowing
matrix
of o r d e r
Wallis
exists
(1976)),
and their p o s s i b l e s I orthogonal
(with some
m-sequence,
that
for
in p a r t i c u l a r ,
4"59=236.
a 6-code
then there
show that n o t a t i o n s
are e q u i v a l e n t
of s u p p l e m e n t a r y
Geramita,
s I ~ s 2 ~.°.~
we will
4t,
supposition
of length m.
3.1.(Turyn,1974) . If there e x i s t
a Baumert-Hall
Later
a 4-symbolical
let us introduce
to show the e x i s t e n c e
Turyn made
of length m and
a Baumert-Hall
were
investigated
applications design
of 6-code
limitations).
let us introduce
array
for type
construction.
and of s u p p l e m e n t a r y To define
the n o t a t i o n
52
X = {{a I
where
ai,j,
}n ,i i=I
i=1,2,...,m,
,{a2 ' }n i i=I
j=1,2,...,n
n '{am,i}i=1}
'
are any real values.
n-j Nx(J)
=
(al ial i+j + a2 ia2,i+j ' '
i=I
+ "'" + am,iam, i+~)j j=1,2,...,n-1
Px(j)
= Nx(j)
where
riodic
B = blb2...bn;
I...aI;
Px(j)
DEFINITION be c a l l e d
3.3
•
x
Introduce
matrices). a periodic
(respectively) {a I , i ~]n i=1
sequences
and a n o n - p e -
of set X [209].
,{a2 ' i }ni = I ' ' ' " {am,i} ~ =I will
of length
n provided
, i=I,2 .... ,m; j = 1 , 2 , . . . , n
some p r o p e r t i e s
PROPERTY
(3.4) (3.5)
of s u p p l e m e n t a r y
sequences
of l e n g t h
(Robin-
1976)
3.3.
Let AI, A2,
..., A m be s u p p l e m e n t a r Y
sequences
of
n. Then
I. A~, A~,. "'' A*i' Ai+1' AI, A2, length
(numbers
= albla2b2...anbn
(j) = 0 , j=I,2 ..... n-1
son and Wallis,
length
functions
supplementary
A/B
will be c a l l e d
~02 ]. Sets
I. ai, j 6{-I.+I} 2. N
objects
and Nx(j)
autocorrelation
-A = - a l - a 2 - . . . - a n ;
AB = ala 2...anbn;
ai,b i are a r b i t r a r y
Functions
(3.3)
+ Nx(n-j) , j=I,2 ..... n-1
A = aiA2..°an; A* = a n a n _
(3.2)
..., Ai,
Ai+2 , "'" A m
-Ai+1,
and
..., -A m are
supplementary
sequences
of
n.
2. AIA2,
AI-A2,...,A2i_IA2i
...,A2i_i/A2i,A2i_i/-A2i 3. If Nx(j) not true.
, A2i_1-A2i,...
and At/A2,
are
supplementary
sequences
= 0 for e v e r y
j, then Px(j)
= 0. R e v e r s e
AI/-A2,...
of length
2n.
statement
is
53
Using ructions
this
properties
for o r t h o g o n a l
x 2 , . . . , x I of w h i c h In p a p e r s tement
was
and
designs
and W a l l i s
of t y p e
do n o t c o i n c i d e
of the
author
and
have
proposed
the c o n s t -
( S l , S 2 , . . . , s I) p a r a m e t e r s
x I,
simultaneously.
Sarukhanian
(1979,
1982)
following
sta-
proved.
STATEMENT sary
Robinson
3.2.
For
sufficient
the e x i s t e n c e
the e x i s t e n c e
of a
~ (t,m)-sequence
of t - s u p p l e m e n t a r y
it is n e c e s -
sequences
of
length
m. NECESSITY. V i of
Let V = {Vi }mi=I
form V i =
j=1,2,...,t
are
(ai,j)tj=1
vectors
Hence,
of
m. T h e n
more
than
and
length
is true.
By
substitution
one
(3.1)
t orthogonal For
sufficient
In fact,
= {a i ,31=I .} m since
the v a l u e s
V of
we have
0 , j : 1,2,...,
3.2 Aj,
j=1,2,...,t
can note
3.1.
Aj
vectors
m-1
j=1,2,...,t
are
supplementa-
m.
Aj,
the c o n d i t i o n
COROLLARY sary
Let
m.
the
that
length
to d e f i n i t i o n
length
us p r o v e
of
(3.1) (3.1)
Let
with
sequences
t E ai,kai+j, k k=1
SUFFICIENCY.
satisfy
in
according
ry s e q u e n c e s
length
so
V i and Vi+ j
m-j E i=I
' i_1,2,...,m.
supplementary
is a 6 ( t , m ) - s e q u e n c e ,
be a 6 ( t , m ) - s e q u e n c e
that
be
vectors
hence,
Vi :
sequence
(-1,+1)-vectors. the e x i s t e n c e
the e x i s t e n c e
supplementary
The
sequences
(ai,j)t:]_
, i=1,2,...,m
~m V = {ViJi= I c o n s i s t s theorem
of
of n o t
is p r o v e d .
of a 6 ( 2 , m ) - s e q u e n c e
of a s u p p l e m e n t a r y
Golay
it is n e c e s sequence
m.
PROPERTY
3.4.
A non-periodic
autocorrelation
function
of the
se-
quence
Q = {a I, a2, .... am,
of c o m m u t a t i v e following
in p a i r s
relations.
bl,
of o b j e c t s
b 2 , . . . , b n]
(vectors,
, m~ n
matrices)
is d e f i n e d
by
of
54
n-j
m-j j Z aiai+ j + Z b.a i=I i=1 i m+i-j
m NQ
=
Z aibi+4_ mJ i=I
+
Z1bibi+ j , i <_ j < m - 1 i=
n-j Z1bibi+ j , m < j < n-1 i=
(3.6)
m+n-j Z aibi+j_ m , n < j < m+n-1 i=I
PROPERTY
3.5.
If there exists a ~(4,n)-sequence,
then there exists
a 6(4,2in)-sequence. PROPERTY
3.6. If there exist 4-supplementary
then there exists a
6(4,2n)-sequence.
LEMMA 3 •I . Let A = be orthogonal thogonai
sequences of length n,
{ai} ~ =I
' B =
{bi} ~ =I
' C = {c i }n i=I' D = {di} ~ =I
n-dimensional
(-1,+1)-vectors.
Then there exist
(n+4t)-dimensional
(-1,+1)-vectors,
where
PROOF.
One can verify that for t =
four or-
t=I,2,...
I abovementioned
vectors become
A I ={ (ai)ni=I , I, I, I, I} ' BI ={(bi)ni=1'-1'1'-1'1}
, (3.7)
C I ={ (ci)ni=I' Further,
to obtain
-I,
I, I, -I}, D1={(di)ni=1,-1,-1,1,1}
four orthogonal
tors we must repeat the construction LEMMA 3.2.
From statement
sional o r t h o g o n a l nal vectors
(-1,+1)-vec-
times. sequences of length n,
sequences
of length n, t=I,2,...
3.2 and lemma 3.1 we have four 4(t+1)-dimen-
(-1,+1)-vectors.
for orthogonal
get a sequence
(3.7)
If there exist 4-supplementary
then there exist 4 ( t + 1 ) - s u p p l e m e n t a r y PROOF.
(n+4t)-dimensional
Now by substitution
4-dimensional
(4t+1)-orthogo-
vectors of ~(4,n)-sequence,
of length n. One can obtain that the sequence
is a supplementary
we
received
4(t+1)-sequence.
T H E O R E M 3.2. For the existence ry and sufficient the existence ...,t, ai,46{0,-1,+1}j
satisfying
of a 6(t,n)-sequence
it is necessa-
n of t vectors X i = (ai,j)j= I , i=1,2, the following conditions:
55
I. X I + X 2 +...+ 2. X
, X
i
= 0,
3
n-j ~ (al,ial i=I
3.
NECESSITY. pj
Xt
i ~ j, i,j
i+j
Let V = {Vi}~= 1
that
piPj
Let
us
= 0,
i ~ j,
the
Prove
that
Xi =
First
two c o n d i t i o n s
+...+
N o w on the
that
= 0,
j=1.2,...,n-1
and
k-dimensional
= 1,2 .... ,t
let pj,
(k ~ n )
(-1,+1)-
t I PixX i=I
on the o t h e r
n-j I i=I
Nv(J)
i=1,2 .... ,t
(3.9)
(3.10)
from
sequence
one h a n d
= k,
±
' i=1,2,...,t
follow
the
a n d p2
V in f o r m
(3.9).
satisfy Verity
the
the
conditions
validity
(3.1).
of c o n d i t i -
V in f o r m
V = {al,iP I + a 2 , i P 2 +...+
=
(3.8)
a t , i + j)
be o r t h o g o n a l
i,j
(ai,j) nj=1
Substitute
Nv(j)
n;
be a 6 ( t , n ) - s e q u e n c e
sequence
V =
sequence,
length
is
substitute
3.
of
= 1,2, .... t
+ a2,ia2,i+j
6{Vi}~= I , j=l,2,...,t,
vectors,
on
(-1,+1)-vector
= 0,
(3.11)
at,iPt}~= I
j=1,2,...,n-1
since
V is a 6 ( t , n ) -
hand
t ~p2 n-j Z a a = k I m=1 m, 3 m , i + 3 m i=I
t ~ a ~a m=1 m,l m , i + j
is n-j
t 7 a .a i=I m=1 m,l m,i+j
SUFFICIENCY. fy the c o n d i t i o n s
Let
vectors
(3.8)
and
= 0,
Xl = (3.10)
j=1,2,...,n-1
(ai,j)n
and
j=1
respectively.
Pi' Form
i=I,2,..
"
,t,
a sequence
satisof v e c -
tors
}n D = { a l , i P I + a 2 , i P 2 +"
"+ a t , i P t
i=I
(3.12)
58
Show
that
D is a
~(t,n)-sequence.
Calculate
n-j t t E ( E a iPm) ( E a i+jPr) i=1 m=1 m, r=1 r,
ND( j ) =
n-j = k E i=I
with
that
and
t E a .a r=1 r,l r,i+j
in v i e w
'
j:1,2, .... n-1
Now any
from
conditions
j, ND(j) Note
reduced and
= 0. The
that
of
re e x i s t s
there
length 3.3.
Let
a 6(t+k,
PROOF. sequences.
and
from
of c o n s t r u c t i o n
a simple
satisfying algorithm
there
exist
6(t,m)-
and
Then
(ai,3)m3 :I
vectors
Xl :
(3.8) , Pi' the
that
for
of
the
$(t,n)-sequence
conditions
for c o n s t r u c t i o n
(3.8) of t h o s e
@(k,n)-sequences.
Then
the-
m+n)-sequence. k and V 2 : E Yi×Qi i=I
with
we h a v e
n6{2,4,6,...,40,42}.
t Let V I = E X Ixpl i=I
the c o n d i t i o n s
(3.13)
is p r o v e d .
of t v e c t o r s
exists
n,
theorem
the p r o b l e m
to c o n s t r u c t i o n
THEOREM
vectors
theorem
firstly,
secondly,
vectors
of the
(3.13)
same
and
i=I,2, .... t, Qi'
dimension
q,
,
~(t,m)-
Yi =
(b i
and
)n ,3 3 =I
i=1,2,...,k,
are
~(k,n)satisfy (-1,+1)-
q >_k + t, and
PiPj
= 0,
i # j,
i,j
= 1,2 ..... t;
QiQj
= 0,
i ~ j,
i,j
= 1,2 ..... k; (3.14)
PiQj
= 0,
i = 1,2,...,t,
p2 2 = Qj = q i Now
f r o m the p r o p e r t y
j = 1,2,...,k;
, i = 1,2,...,t
3.4 we h a v e
, j = 1,2,...,k
that
m
V = { (al,iP I + a 2 , i P 2
+...+
at,iPt) i: I , (bl,iQ 1 + b 2 , i Q 2 + . . . +
+ b k , i Q k ) ~ : I] is a
6(t+k,
m+n)-sequence.
COROLLARY ence,
where
3.2.
There
exists
a i are n o n - n e g a t i v e
a 6(4,2
aI
integers.
10a226a3+2a410a526a6)-sequNote
that
a @-sequence
with
57
these
parameters
+I)),
6 ( 4 , 2 a I (10 a 2 + I ) 2
THEOREM
includes
3.4.
If
in p a r t i c u l a r ,
6a3)
there
6(4,2
aI
a3
I0a2(26
+
al +I)I oa226a3) .
, 6(4,(2 exists
sequences
a
d(4,n)-sequence,
then
there
exist
T(4,n,In)-matrices. PROOF. we
have
4 = E V.×X. i:I l I
V
that
the
=
( a i ) ni=I
Xl
satisfy
Let
the
TI
be
a
6(4,n)-sequence.
, X2
conditions
n ui_1 = Z a. i=I i
=
( b i ) ni=I
(3.8).
, X3
Verify X
=
This
theorem
' X4
=
these
matrices
n ui_1 = Z b. i=I l
,
T2
'
T 4 = Z d U i-I i= I i
circulant
only
that
3.2
(di)n : I
, (3.15)
n
5th
{XI,X2,X3,X4}. means
( c i ) in: 1
:
Introduce
n
that
the
vectors
T 3 = E c U i-I i= I 1
Show
From
matrices
condition One
can
in
show
non-diagonal
are
T(4,n,In)-matrices.
definition that
for
elements
of
of
T-matrices.
every
j,
Nx(j)
Denote
= 0 and
Px(J)=0.
matrix
4
E i=I are
0 and
diagonal
elements
T. T T
i l
(according
to
(3.8))
are
n,
that
is
reduce
to
4
i:I
The
theorem Thus,
struction Now and
is p r o v e d . the of
we
construction a
square
vectors
of
a Baumert-Hall
array
we
the
con-
6-sequence.
consider
investigated Let
T.TT = nl I i n
by
the
generalization
the
author
and
Vi(XI,X2,...,Xn)
(-1,+1)-matrices
of
order
of
notation
Sarukhanian
, i=1,2,...,t k,
satisfy
"@-code"
in p a p e r s coordinates
the
conditions
introduced
(1979, of
1981).
which
are
58
Vi VT
= 0,
i ~ j,
i,j
= 1,2 ..... t (3.16)
n
v.vT 33
where
vT
is a r o w - v e c t o r
STATEMENT tors
= Z x.xT i=i I I
3.3.
If
, j = 1,2 ..... t
of f o r m V Ti(xI"T
(A,B,C,D,Ik,k)
VI(A,B,C,D) , V2(-B,A,-D,C)
the c o n d i t i o n s DEFINITION element
,Xn)T
is a W i l l i a m s o n
, V3(-C,D,A,-B)
family,
then
, V4(-D,-C,B,A)
vec-
satisfy
(3.16)• 3.4.
of w h i c h
neralized
X2,T
6-code
A square
matrix
Q ( V I , V 2 , . . . , V t)
is of f o r m V i or -V i w i l l of
length
of o r d e r
be c a l l e d
m = s1+s2+...+st
with
m each
a t-symbolical
density
k, b a s e
ge-
n pro-
vided
Q(Vl' where
1 ' VT2 ' ' "
V2'''''Vt)QT(v
s i is the n u m b e r
of a p p e a r a n c e s
of m a t r i x
Q(Vi,V2,...,Vt)
cods
6(t,m,k,n).
by
If Q ( V i , V 2 , . . . , V t ) be c a l l e d
a circulant
STATEMENT exists
. Let
VI =
is the c i r c u l a n t generalized
Let
there
Consider
Q = V1×
ai,
I,
I,
I)
(3.17)
V i in any
t-symbolical
matrix,
row
(column)
generalized
then
6(t,m,k,n)
6-
will
6-code.
exists
generalized 4 Let V = Z V xX be i=I i 1
(I,
of v e c t o r
us d e n o t e
a circulant
PROOF.
where
3.4.
t = i=1 Z s .1 v .1v T1 xI m
.Vk)
a 6(4,m)-sequence.
Then
there
6(4,m,k,n)-code. a
6(4,m)-sequence
, V2 =
(-I,
1, -I,
I)
V4 =
(-I,
-I,
I)
I,
with
, V3 =
(-I,
I,
I, -I)
,
the m a t r i x
m£ a .ui_ I + mZ b U i-I i=I i V2xi= I i
+
bi,
are
ci,
di,
X2,X3,X 4 respectively•
i=1,2,...,m, According
V3x
~ ciUi - I + V4xim d.Ui-1 i=I i=I i
coordinates
to t h e o r e m
3.2
of the v e c t o r s
they
satisfy
the
'
X I, condi-
59
tions
(3.8).
(3.17)
It is e a s y
hence,
this matrix
It is a l s o e a s y fact,
first
p =
to s h o w t h a t m a t r i x
Q satisfies
the c o n d i t i o n
is 6 ( 4 , m , k , 4 ) - c o d e
to s h o w t h a t
r o w of t h e c i r c u l a n t
the r e v e r s e matrix
statement
is n o t
true.
In
P,
VI×(I+U 5) + V2x(U+U 7) + V3×(U2+U3+U4-U6 ) + V4×U8
of o r d e r
9 is n o t a ~ ( 4 , 9 ) - s e q u e n c e ,
though
the m a t r i x
P is a 6 ( 4 , 9 ,
k,4)-code. THEOREM k,4)-code
3.5.
For
the e x i s t e n c e
it is n e c e s s a r y
T-matrices
of o r d e r
NECESSITY.
and
of a c i r c u l a n t
sufficient
generalized
the e x i s t e n c e
~(4,m,
of c i r c u l a n t
m.
Let Q(VI,V2,V3,V4)
be a c i r c u l a n t
generalized
~(4,m,k,
4)-code
Q ( V I, V 2, V3,
L e t us p r o v e
that
4 = ~ V × x i=I 1 l
V 4)
X. are i
T-matrices
of o r d e r
m.
The conditions
X i , X j = 0,
i ~ j, X i X j = X Xi, i , j = I , 2 , 3 , 4 , a r e t r i v i a l . It is e a s y to 4 3 verify that Z X. is a ( - 1 , + 1 ) - m a t r i x of o r d e r m. T a k i n g i n t o a c c o u n t i=I i t h a t V V. = 0, i ~ j, c a l c u l a t e 13
ooT From
the
definition
SUFFICIENCY It a l s o
holds
THEOREM
3.6.
exists
and
Let
i=1
xix v v
3.4 we h a v e theorem
there
4 I X xT = mI i= I 1 i m
is o b v i o u s .
exist
T-matrices
of o r d e r
m.
Then
there
a 6(4,m,k,4)-code.
STATEMENT n)
of the
4
3.5.
Let
there
exist
(A2,B2,C2,D2,Rk,k) . Then
Williamson
there
exists
families
( A I , B I , C I , D I , I n,
a generalized
6(2,2n,n,4)-
code. COROLLARY
3.3.
There
exists
a symmetric
Williamson
family
60
(A,B,C,D,I
i , n m l) , w h e r e
n 6 LI U L2
,
nm m 6 {14,
STATEMENT code
and
3.6.
a Williamson
family
amson
Let
26,
30,
there
38,
exist
family
42,
50,
54}
a generalized
(A,B,C,D,In,n).
circulant
Then
there
~(4,m,k,4)-
exists
a Willi-
(X,Y,Z,W,RmXIn,mn) . 4
In f a c t ,
let
~(4,m,k,4)-code.
Q(VI,V2,V3,V4)
= E Vi×Xz i:I verify that
Then one can
be
a circulant
generalized
matrices
X = AxX I + BxX 2 + CxX 3 + DxX 4 Y =-BxX I + A×X 2 - DxX 3 + CxX 4 Z =-CxX I + DxX 2 + A×X 3 - BxX 4 W =-D×X 1 - CxX 2 + BxX 3 + AxX 4
form
the
family
THEOREM de and
3.7.
PROOF.
type
Let
@(2,m,k,4)-code order
4t;
VI =
where
there
A ( V I , V 2) and
let
suppose
can
also Let
verify
array
of o r d e r
order
4 t m i,
= VlXK I + V2xK 2 H(AI,A2,A3,A4)
that
generalized
the
vectors
,
V2 :
4t.
be be
of
type
a
array
form
T T T T (-Yo,Xo,-W0,Zo)
(3.18)
family.
the
theorem
3.5 K I and
yT i_ixK2
exists
generalized
a Goethals-Seidel
VI,V 2 satisfy
_
there
a circulant
VI,V 2 are
is a W i l l i a m s o n
following
Then
6(2,m,k,4)-co-
i=I,2,...
vectors
consider
Xi_IK I
a circulant
that
from
=
Xi
of
(Xo,Yo,Zo,Wo)
that
us
exist
type
array
(Xo,Yo,Zo,Wo,Rk,k)
One Note
Let
a Goethals-Seidel
Goethals-Seidel
of
desired.
the
K 2 are
conditions
(3.16).
T(2,n,Bn)-matrices.
matrices:
,
Yi
= Yi-lXK1
T + Xi-lXK2
' (3.19)
T Z i = Zi_IK I - Wi_ixK 2
,
Wi
= Wi-lXK1
+ zTi-lXK2
61
Let us p r o v e
that m a t r i c e s
pQ = Qp
Xi,
P(BkXBn)QT
Yi' Wi'
Zi s a t i s f y
the c o n d i t i o n s
= Q(BkXBn)PT
(3.20) P'Q6{Xi'
Verify
the c o n d i t i o n
(3.20)
Yi'
Zi' Wi}
for
i=I.
"
2 T T TT 2 XIY I = X ~ Y o x K 1 + X o Y o x K i K 2 - Y o y o x K 2 K I - Y o X o x K 2 YiX1
= y o X o x K 2I _ y o y oT× K i K 2 + X ~ X o x K 2 K 1 - X oT Y oTX K 2
Taking that
,
into a c c o u n t
the m a t r i c e s
we have
the c i r c u l a r i t y
Xo,Yo,Zo,Wo
satisfy
of m a t r i c e s
the a n a l o g o u s
K 1, K 2 and the fact conditions
(3.20),
XIY I = YIXI . F u r t h e r
T T T x T T T T X I ( B k X B n ) Y I = X o B k Y o X K I B n K I + XoBnXo K I B n K 2 - Y o B k Y o X K 2 B n K I Y~BkXo xK2BnK ~ Y I ( B k X B n ) X I T = Y o B k X oTX K I B n K I T - YoBkYoX K I B n K ~
+ X oT B k X oTX K 2 B n K TI
XT kYOX 2BnK By c o m b i n a t i o n One can
of these
show a n a l o g o u s l y
relations
we have
that c o n d i t i o n s
XI (BkXBn)Y~ (3.20)
are
= YI(Bk×Bn)XI
satisfied
T
for i > I
too. T + K2K ~ = nI n we have F r o m KIK I Xi IxT + yi Y~l + Zi 1zT + W l wT1 = n(XoXT
+ YoY~ + ZoZ~ + WOWS) xI i
(3.21)
n Hence, Seidel
the m a t r i x
type a r r a y
of o r d e r
has one of the forms iPB,~pTB From
appears
(3.21)
H(Xi,Yi,Zi,Wi)
4tn I. In fact,
~P,JPT,!pB,jpTB,
in e a c h
= H] (Xo,Yo,Zo,Wo)
row
(column)
P6{Xo,Yo,Zo,Wo}; of m a t r i x
we have
H]H TI = 4 t n i ( x o x T
+ YoY~
each element
+ WoW~)xI 4tn I
is a G o e t h a l s of m a t r i x element
H I precisely
HI
+p,+pT,
tn i times.
62
COROLLARY
3.4.
4n i, 8n ~, 40n l,
There
exists
104n l, 8kn l,
a Goethals-Seidel
16kn ~,
type
104kn i, w h e r e
array
k E L 1U L 2
of o r d e r ,n=2,10,26,
i=I,2,... n B ={ b i } i = 1 ,
L e t A = {a i} =I'
~n-t+1 ~-~+I C ={ c i ~ i = 1 , D ={ d i} =
be
(-1,+1)-sequences. DEFINITION ences
3.5.
A,B,C,D
will
be c a l l e d
supplementary
Q(n,t)-sequ-
provided
n-t-j+1 X (aiai+j~ i=I
b i b i + j) = 0
+
b i b i + j~
,
+
C .iC ~i j + d . di i+j
j = 1,2,...,
+
n-j X i = n - t - j + 1 (aiai+j
+
n-t
(3.22)
n-j j = n-t+1,...,
X (aiai+ + bibi+ ) = 0 i=I J J
n-1
'
NOTE
3.2.
For
t = I A,B,C,D
are
NOTE
3.3.
For
t = 2 the n o t a t i o n
supplementary
sequences
of
length
n.
that
of T u r y n In fact,
sequence
for t = 2 the c o n d i t i o n s
alan
tions
that
(3.23)
symmetry Now
of
Turyn with
are
Turyn
+bb
=0
nn-j
A and C
introduce
Turyn
chine
processing
n = 2
: X ={ (+-) , (++) , (+) , (+) }
(3.23)
sequences.
introduced
supplementary
in
with
become
+ cici+ j + d 1d1 +) 3+ a a n n - j
himself
sequences
let us
(3.22)
+ blb n = 0 , j=1,2,...,n-2
for t = 2 A , B , C , D Note
coincides
[203].
n-j-1 Z (aiai+ j + b.b. i=I i l+j
So,
of Q ( n , 2 ) - s e q u e n c e
sequences
requirements
satisfying
namely,
the c o n d i -
symmetry
and
skew-
[203 ]. sequences
[203 ].
n = 3 : X ={ (+++) , (++-) , (+-) , (+-) }
of
length
n, o b t a i n e d
by ma-
63
n : 4
: X ={(++--),(++-+),)+++),(+-+)}
n : 5
: X :{ (++-++) , (++++-) , (++--) , (+-+-) }
n = 6
: X ={ ( + + + - - - ) , (++-+-+) , (++-++) , (++-++I }
n = 7
: X ={ ( + + + - + + + ) , ( + + - - - + - ) , ( + + - + - - ) , ( + + - + - - ) }
n = 8
: X ={ ( + + - + - + - - ) , ( + + + + - - - + ) , ( + + + - + + + ) , ( + - - + - - + ) }
n =
13:
X ={(++++-+-+-++++)
, (+++--+-++-++-)
(+++--+-++---) n =
(+ . . . . + - + - + ÷ + + - )
a
3.8.
Let
,
}
15: X ={ ( + + - + + + - + - + + + - + + )
THEOREM
,(+++-++--+---)
there
, (+++-++---++-++-)
, (++++--+-++ .... ) ,
}
exist
Q(n,t)-sequences.
Then
there
exists
6 (4,2n-t+1) -sequence. PROOF.
Prove
that
. ; . ni=I -t+1 V ={ ( c i , - c i , ~ i., - ( l i is a
6(4,2n-t+1)-sequence.
Nv(j)
2
= 2
From
' ( a i , a i , b i , b i ) n =I }
property
n - t + 1 -j Z ( C i C i + j + d i d i + j) i=1
3.4
for
I < j
we
have
n-j + 2 Z (aiai+ j + bibi+j) i=I
n-t+1-j E ( a i a i + j + b i b i + j + c i c i + j + d i d i + j) i=I
+ 2
=
n-j E (a~a~ i=n-t+2-j i 1+3
bibi+ j )
By
(3.22)
we
get
NV(j)
For
j=n-t+1,...,n-1
Nv(j)
For
= 0,
j = 1,2 .... n - t
from
(3.22)
we
find
n-j = 2 k=1 ( a k a k + J + b k b k + j ) = 0
j=n,n+1,...,2n-t
we
have
by
analogy
Nv(j)
= 0.
The
theorem
is
proved. COROLLARY
3.5.
If t h e r e
exist
four
supplementary
sequences
of
64
l e n g t h n, t h e n t h e r e COROLLARY
3.6.
exists
a 6(4,2n)-sequence.
If t h e r e e x i s t T u r y n
sequences
of l e n g t h n, t h e n t h e r e
exists a 6(4,2n-1)-sequence. COROLLARY
3.7.
For t = n
n+1 h e n c e , { a i , a i , b i , b i } ~ = I
V is a 3 - s y m b o l i c a l is a 2 - s y m b o l i c a l
6 - c o d e of l e n g t h
6-code of l e n g t h n, that
is
n
A = {ai}i= 1
Are
supplementary COROLLARY
Golay
3.8.
B = {bi }ni=i
•
sequences
of l e n g t h n.
If there e x i s t s
a Q(n,t)-sequence,
then t h e r e e x i s t s
a 6(4,21(2n-t+1))-sequence. N o w let us g e n e r a l i z e
the n o t a t i o n
of s u p p l e m e n t a r y
sequences.
Con-
s i d e r a set }n X ={{At, i i=I
}n '
w h e r e Ai, j ( - 1 , + 1 ) - m a t r i c e s
.AT Ai, 3 r,l DEFINITION
3.6.
{A2
,i
i=I'
''''{Am,
i
}n i:1 }
of o r d e r k s a t i s f y i n g
T = Ar,iAi,j
, i,r=1,2,...,m,
the c o n d i t i o n s
j,l=1,2,...,n
[ 91]. A set X w i l l be c a l l e d
.
a D(m,n,k)-sequence
provided n-j m Z i A .A T i=I r=1 r,1 r , i + j NOTE
3.4. A D ( m , n , 1 ) - s e q u e n c e
NOTE
3.5.
= 0 , j=1,2,...,n-1
coincides
If X is a D ( m , n , k ) - s e q u e n c e ,
with a 6(m,n)-sequence. then f o l l o w i n g
relation
is
true:
m j AT E ( ~ Ar,i r,n+i-j r=1 i=I
+
n-JA AT ~ i r,i+j i=I r,
j=1 ,2 , . . . ,n-1
)
=
0
(3.24)
65 THEOREM
3.9.
Let
there
exist
a Williamson
family
(A,B,C,D,Ik,k)
Then there exists a D(4,n,k)-sequence. 4 L e t V = E V.×X. be a c ( 4 , n ) - s e q u e n c e . Then from i=I i 1
and
a o(4,n)-sequence. PROOF.
theorem
3.2
the v e c t o r s
Xl
satisfy
=
(ai)ni=I
(bi) n :I
' X2 =
the c o n d i t i o n s
=
B
(3.10).
n
X = { { A i } ~ I ' { i}i:1
n (ci)i=1
' x3 =
L e t us c o n s i d e r
C
n
'
Bi
,
' { i}i=1
{Di}
~
' X4
= (di)
=I
a set
=I
}
where - b l•B l
Ai = aiA C
=
Prove
a.C 1
-
that
,AT A 1 l+j
b.D 1
-
+
ciC
c.A l
+
- diD d.B 1
,
D. l
X is a D ( 4 , n , k ) - s e q u e n c e .
= aiai+3
,AA T
-
=
aiB
+
+ biA
a.D l
+
b.C l
For
this
-
c .1D
c.B l
-
+
purpose
d •lC
,
d.A 1
calculate
a b .AB T a.c, ,AC T a.d .AD T biai+jBAT+ i i+ 3 - i l+ 3 - i i+ 3 -
+ b.b. 1
+ c i b lT+ j .C B
=
.BB T + b.c.
l+ 3
1
.BC T + b,d.
l+ 3
+ cici+jCC T + cidi+dCDT
1
.BD T - c.a.
l+ 3
l
.cAT+
l+ 3
- d.a.1l+ 3"DAT + d~bl i+ 3"DBT+
+ dici+jDC T + didi+jDDT .BT B1 l+j
.a ,BB T + a,b. .BA T + BD T - a . d .BC T + b,a. .AB T+ = al i+ 3 i i+ 3 aici+j 1 i+ 3 1 l+] + blbl+jAAT. .
+ bici+jAD T - bidi+jAcT
+ cici+ j DDT-
+ c i a i + j D B T + c i b i + 3.DAT+
c i d i + j D C T - d a. .CB T - d . b .CA T - d . c cDT+ i l+ 3 l l+ 3 i l+ 3
+ didi+jccT CiC~+ j = aiai+jCC T - aibi+jCD T + aici+jcAT + b.b. 1
.DD T - b.c.
l+ 3
1
DA T - b d
l+ 3
+ cici+jAA T + cidi+jABT
i
+ aidi+jcBT
DB T + c a l+ 3
+ diai+jBcT
1
- biai+jDcT+
.AC T - c.b. l+ 3
- dibi+jBDT
l
.ADT+
l+ 3
+ d i c i + 3.BA T+
+ didi+jBBT D.DT+.I 1 3 =
aiai+j DDT
+ aibi+jDcT
- aici+j DBT
+ aidi+3'DAT
+ biai+jcDT+
66
-
+ bibi+jccT
bi
c
C B T + b d~ ~CA T c a ~BD T - c b. .BC T+ l+ 3 i l+ 3 1 l+ 3 i l+ 3
- cidi+jBAT
+ cici+jBBT
+ diai+jADT
+ d .ib i+ 3.AC T - d i c i + j A B T+
+ didi+jAAT
By
summation
of e q u a t i o n s
obtained
we h a v e
n-j
z
{AAT + iB[ +j + C C1 i i+3
i=I
l+j
+ D DT+ ) = i i 3
(AA T + B B T + C C T + D D T)
n-j i=I
(aiai+ j + b i b i +
Since the
sum
the v e c t o r s
(3.25)
theorem
3.9.
i=1,2,3,4,
Hence,
3.6 a n d
the
satisfy
the c o n d i t i o n s
(3.8)
set X is a D ( 4 , n , k ) - s e q u e n c e .
exists
, The
theorem
3.10.
If
exists
3.9
follows:
a D(4,n,k)-sequence,
then
there
too. there
exists
a Turyn-sequence
of
length
n
, then
a D(4,2n-l,k)-sequence.
In fact, the
from theorem
If t h e r e
a o(4,n)-sequence
COROLLARY
then
is zero.
corollary
COROLLARY
there
Xi,
(3.25)
) , i=I,2,...,n-I + d d i i+j
is p r o v e d .
From
exists
+ c.c i l+j
j
if we h a v e
statement
instead
of c o r o l l a r y
of A , B , C , D 3.10
one
matrices
can prove
of o r d e r
k +I or
analogously
-I,
to t h e
3.9.
Plotkin
a p p r o a c h . The
approach
is b a s e d
on t h e n o t a t i o n
A square
G(Xl,X2,...,Xm)-matrix
of m a t r i x
decomposition. DEFINITION each
element
composition
3.7.[192].
of w h i c h
is of
m
3.6.
n. H a d a m a r d
l
or -x. w i l l l
be c a l l e d
n
a D(n,m)-de-
provided
GG T = ~ Z m. i=I NOTE
form x
of o r d e r
(3.26)
in
I. G ( t 1 , ~ 1 , . . . , t l ) - m a t r i x
matrix
decomposition.
x~ I
G(I,1,...,I)
is an H a d a m a r d
let us c a l l
a matrix
matrix
generated
of o r d e r D(n,m)-
67
2. The of o r d e r
special
3. M a t r i x Baumert damard In tion
of D ( n , m
)-decomposition
are
Baumert-Hall
array
n for m=4.
Williamson
-
cases
array
satisfying
and Hall matrices [192]
for m=1.
in
the c o n d i t i o n
1965;
in p a r t i c u l a r ,
of o r d e r
M.Plotkin
(3.26)
was
this
constructed
allows
first
to c o n s t r u c t e d
by Ha-
156.
proposed
two h y p o t h e s e s
about
D(n,m)-decomposi-
namely, I. For
any natural
number
k there
exist
2. To a n y H a d a m a r d
matrix
H of o r d e r
D(4k,4)-
and D(8k,8)-decom-
position.
l)-decomposition
G(Xl,X2,...,Xl)
N o w we w i l l a convenient
for
idea of P l o t k i n THEOREM m>
I
and
one
give
Plotkin
further
(1972)
that
1=4,8
corresponds
G(I,I,...,1)=H.
and Baumert-Hall
considerations
a D(kl,
form.
These
[ 295] t h e o r e m s theorems
reveal
in the
approach.
3.10.
(Plotkin,
can c o n s t r u c t
1972).
For
its g e n e r a t i n g
any
Hadamard
matrix
D(m,2)-decomposition
of o r d e r and
m,
D(2m,4)-
D(4m,8)-decompositions. THEOREM
Ik,k)
and
order
nk.
3.11.
Let
there
following
DEFINITION
3.8
A set of
3. K i K ~
= ym Im
, i=1,2, ... ,i
4. K . K ~ 13
+ K.K~ 31
= 0,
(-1,+1)-matrix
an H a d a m a r d
square
(0,-1,+1)-matrices
of o r d e r
m;
i ~ j, ij=1,2 ..... 1
a 1-elemental 3.9.
exists
i,j=1,2,...,l;
is a
DEFINITION
there
( A I , A 2 , . ° . , A m,
the c o n d i t i o n s
1 2. Z K. i=1 i
be c a l l e d
Then
family
"
m satisfying
I. K. * K = 0, i j
a Williamson
matrix
definitions.
[15] "
of o r d e r
exist
D(n,m)-decomposition.
Introduce
will
such
kl,
frame
[15 ]. A set of
of o r d e r square
m. (0,-1,+1)-matrices
{Ki}l i=I
of
68
i
{U}i=l
of o r d e r
1. Ui
2.
will
* H~ ~ 0,
1 Z H~ i=I l
= ~I 1 m
4. n i n
÷
that
W(m,m/l)
the c o n d i t i o n s
i , j = 1 , 2 ..... 1
(-1,+1)-matrix
semi-frame
elements
of a f r a m e
and that
in some
with
that
cases
PROBLEM n~p(m)
problems I. For
construct
are w e i g h i n g
notation
unities
set
matrices
of 4 - e l e m e n t a l
of
frame
(Turyn,1974).
1 = I x H i=I i l
a U(m,l)-semi-decomposition.
is a 1 - e l e m e n t a l
Following
the
m.
[15 ]. A m a t r i x
U(Xl,X2,...,Xll..
{Ui}~= 1
of o r d e r
(semi-frame)
of B a u m e r t - H a l l
3.10.
be c a l l e d
m;
j, i , j : 1 , 2 ..... 1.
a 1-elemental
DEFINITION
will
of o r d e r
' i=1,2,...,i;
0, i
be c a l l e d
coincides
i ~ j,
is a
3. H i U
Note type
m satisfying
semi-frame
if the
of o r d e r
set of m a t r i c e s
m.
appear.
arbitrary
natural
numbers
m,n,k,
a D(4mn/k,n)-deconposition
4mn~0(mod
4)
,
(H(4n~/k,n)-semi-decompo-
sition. PROBLEM numbers on
2. For e v e r y
n,k,n~p(m),
4mk/k~0(mod
matrix
of o r d e r
4) c o n s t r u c t
4m a n d
for a r b i t r a r y
a D(4mn/k,n)-decompositi-
(U(4mn/k,n)-semi-decomposition. Note
that
hypotheses
problems
and that
4m is the o r d e r has
Hadamard
frame)
solution
coincide
I has
for k=4,
an a f f i r m a t i v e
solution
D(4m,4)-decomposition, with
regard
n=4,8
wich
Plotkin
for k = I , 2 , 4 ,
whereas
problem
2
to H ( 4 m n % k , n ) - d e c o m p o s i t i o n
k=I,2,4.
L e t us give Let
problem
of e x i s t i n g
an a f f i r m a t i v e
for n=4,
introduced
some
properties
a set of m a t r i x of o r d e r
m. T h e n
1 {Ki}i=1 the
of
1-elemental
1 ({Hi}i=1)
following
frames
and
is a 1 - e l e m e n t a l
statements
are
true.
semi-frames. frame(semi-
69
STATEMENT I. A s e t mi-frame) 2.
•
frame)
{liKi}~=1
of order
1 Z
I.K.
i= I
3
3.7.
1
A set
, i. = +I
1
1
mk,
is a n H a d a m e r d
({Hi xH}l i=I
'
where
element
of t w o H a d a m a r d
frame
(se-
matrix
of o r d e r
m;
--
{Ki×H} ~ I
of o r d e r
, i i = _+I is a 1 - e l e m e n t a l
m;
=
4• E a c h
({liHi}~=1
is a 1 - e l e m e n t a l
H is an H a d a m e r d
of the
matrices,
frame that
1 {Ki}i= I
is K
matrix
frame
of o r d e r
(semi-frak;
c a n be r e p r e s e n t e d
= H + H'
as t h e
sum
;
1
5. F o r a f r a m e
and
semi-frame m ~ 0
6.
1 Z i=I
1 .x K i i 1
s i g n of o r d e r LEMMA elemental
LEMMA
Let
frame
of
PROOF.
(mod 4)
(m/l,m/l,...,m/l)
1
= +I
, orthogonal
de-
1
there
exist
order
p.
frame
3.4.
can construct
m holds
m.
3.3.
a l-elemental
is a t y p e
of o r d e r
There
a 1-elemental
Then
of o r d e r exists
for
any
natural
Hadamard
D(12,1)-decompositions
number
matrix
with
an H a d a m a r d
all
matrix
of
of
+ + + - - + + + - - + + - - +
+ + + + + - - - - + + - - + +
- - + - - + + + - - + - - + - - +
12
=
+ - - - - + + + +
- - - - + + + + - - - - + + + - -
- - - - + + - - + + + + - - - - +
- - + - - - - + + + + + - - + - -
+ - - - - + + - - + + + + - - - -
+ - - + - - + + - - + + +
- - + - - + - - - - + - - + + + +
+
and a 1 l
g there
. . . .
+ - - + + + + +
order
12
1,1~p(12).
+ + + + - - + + + - - + + - -
S
of o r d e r
exists
mp q an
L e t us c o n s i d e r
frame
+ +
order
12
for
which
one
70
Obviously
matrix
theorem
3.10
position
is
exists
the
D(12,1)-decomposition.
D(12,2)-decomposition.
in
[39].
Let
us
b
-c
a
c
b
-a
c
a
b
-c
a
b
b
-a
c
b
c
a
a
b
-c
-a
c
b
-b
c
a
a
b
c -c
b
-a
c
a
-b
c
a
b
b
-a
-c
-a
b
c
a
-b
c
b
c
a
-a
-c
b
b
c
-a
-c
-b
a
c -b
a
a
b
c
-b
-a
c
-b
a
-c
-b
a
c
c
a
b
-a
c
-b
a
-c
-b
a
c
-b
b
c
a
c
-b
-a
-c
-a
b
-c
a
b
b
a -c
a
b
c
-a
b
-c
a
b
-c
a
-c
b
c
a
b
b
-c
-a
b
-c
a -c
b
a
b
c
a
Let
there
an
Hadamard
D(2m i+1,2)-, Let
H2m
be
in
form
represented
H2m Xl,
X 2 is a
X I + X2
is
a
=
a
-b
a -b
c
the
c
3.12.
c
give
b
exist
where
the
to
a
PROOF. be
there
corresponds
introduced
THEOREM there
H12
-b
exists
c
a
c -a
b
The
the
D(12,4)-decom-
D(12,3)-decomposition.
matrix
of
order
2m.
Then
D(4m i+1,4) , D(Sm i+1,8)-decompositiOns.
an
Hadamard
(+
+)xX I +
matrix
(+-)xX
of
order
2m.
2
Obviously
it
can
(3.27)
(-1,+1)-matrix
of
order
2m~m.
(-1,+1)-matrix
of
order
2mxm,
X l X T] + X2X T2 = m I m Obviously
From
(3.28)
matrix
(a,b) xX I +
(3.29)
(-b,a) xX 2
is a D ( 2 m , 2 ) - d e c o m p o s i t i o n . On
the
other
hand
H2m
H2m
can
=H1x(~)
be
represented
+ H2x(10)
in
form
(3.30)
71
where
HI,
H 2 are
(mx2m)-dimensional
T T 0 HIH 2 = H2H I =
matrices
satisfying
the c o n d i t i o n s
,
HIH ~ = H2H T2 = 2 m i m N O W one der
2m 2. By a n a l o g o u s
2m i+I,
i+I)
Then
that
H l X X I + H 2 x X 2 is an H a d a m a r d
constructions
we get
an H a d a m a r d
matrix
matrix
of or-
of o r d e r
i=1,2,...
Writing xm
can v e r i f y
(3.31)
the m a t r i x
-dimensional
obtained
matrices
the r e p r e s e n t a t i o n It is e a s y
A1 =
(a,b,c,d)
(3.27)
Xl,X 2 satisfying
(3.29)
to v e r i f y
in f o r m
that
will
we w i l l
find
(2mi+Ix
the c o n d i t i o n s
(3.28)
be a D ( 2 m i + 1 , 2 ) - d e c o m p o s i t i o n .
the m a t r i c e s
xll x111 + (-b,a,-d,c) x II X2oIf+ (-c,d,a,-b)xll oXl li: 0
+
A2 =
are
(-d.-c,b,a)x
(a,b,c,d,e,f,g,h)
x
l[xl 0 0 0
+
(-c,-d,a,b,g,h,-e,-f)x
+
(-e,-f,-g,-h,a,b,c,d)
+
(-g,h,e,-f,-c,d,a,-b)x
D ( 4 m i + 1 , 4 ) - and The
theorem
COROLLARY (theorem
Then
+ (-b,a,d,-c,f,-e,-h,g)
xI 0 0
I°o x
0 X 01
+
+
x
IJx2 00 0
(-d,c,-b,a,h,-g,f,-e)
(-f,e,-h,g,-b,a,-d,c)
10110 0 XI
+
(-h,-g,f,e,-d,-c,b,a)
x
x
x
X2 0 0
+
0 X 02
+
I° oj
001
0 X2
.
D(8mi+l,8)-decompositions.
is p r o v e d .
3.11.
For
i=0
Let
2nit
theorem
3.12
coincides
with
Plotkin
theorem
3.10).
COROLLARY ces.
0 IIX211 ;
3.12.
for any n a t u r a l
i=1,2,...,r
number
k there
are
orders
exist
of H a d a m a r d
matri-
72
D ( 2 n l i + l n 2" D(Snli+ln~
THEOREM
..'ni,2) -, D ( 4 n
•
r
" ...
3.13.
Let
Let
{X,Y}
there
exist
i . n2 "'"
i " nr'
Then
a 2-elemental there
be a h y p e r f r a m e
D(n,m)-decomposition;
P2i-1,2
i+I
4)-
'
n i ,8)-decompositions. r
and a D(n,m)-decomposition. PROOF.
I
suppose
that
= I, P 2 i , 2 i - I
exists
a D(n,k,m)
of o r d e r
a matrix
= -I
hyperframe
of o r d e r
k
- decomposition.
k, G ( X l , X 2 , . . . , x m) be a
P =
(Pi,j) ni,j=1
is of f o r m
, i=1,2,...,n/2 (3.32)
Pi,j
One
can
= 0
verify
for
i # n/2-I
, j ~ n/2
or
i # n/2,
j ~ n/2-I
that pT
= _p
, ppT
= I n
It is e a s y
to p r o v e
that
S ( X l , X 2 , . . . , x m)
= X x G(Xl,X2,...,Xm)
is a D ( n k , m ) - d e c o m p o s i t i o n . COROLLARY LEMMA there
3.13. There
3.5.
exist
D(mn,1)-
In fact,
one
G2(1,1,...,I) ons,
If t h e r e
The
theorem
exists
exist
+ YxG(Xl,X2,...,Xm)
D(n,l)-
P
is p r o v e d .
a D(8nk,4)-decomposition and
where
n,k6L.
D(m,n)-decomposition,
then
and D(mn,k)-decomposition.
can verify
that
GI(1,1,...,I)
× G 2 ( X l , X 2 , . . . , x I) are
D(mn,k)-
× G(Xl,X2,...,Xk), and
D(mn,l)-decompositi-
respectively. N o w we w i l l
form namely, exists
consider
son type
second
for any W i l l i a m s o n
a generating
The
not
the m a t r i x
author
with
Hadamard
to c o n s t r u c t
be c o n s t r u c t e d
Plotkin type
hypothesis
Hadamard
in some
matrix
altered
of o r d e r
4n t h e r e
D(4m,4)-decomposition. Sarukhanian
matrix
(1982,1983)
of o r d e r
its g e n e r a t i n g
by W i l l i a m s o n
A = I + U + U2 ,
have
proposed
12 a c o n s t r u c t i o n
for w h i c h
D(12,4)-decomposition.
type
matrices
of
for W i l l i a m -
form
B = C = D = i - U - U2
This
one
matrix
can can
73
Now
let u s p r o v e
kin hypothesis THEOREM
about
3.14.
can construct PROOF.
a theorem
H I
=
the validity
Hadamard
matrix
8m. I
~
prove
the
Introduce
theorem
following
E(I,I,I,I,I,I,I,I)H
of order
ml,
1=2,4,8
one
for
1=8.
Let
H be an Hadamard
mat-
,
, ,
U 4 = 1 E(I,I,I,I,I,I,I,I)H
,
=
Plot-
matrices
H 3 = ~ E(I,I,I,I,I,I,I,I)H
5
second weak
H(ml,l)-decompositions.
~2 = ~ E ( I , I , I , I , I , I , I , I ) H I
H
of
H(m,k)-semi-decomposition.
For any
We will
r i x of o r d e r
showing
I
~ E(I,I,I,I,I,I,I,I)H
,
E(I,I,I,I,I,I,I,I)H
,
I H 7 = ~ E|XpI~I~I,I,I,I,I)H
,
H8 = ~ E ( I , I , I , I , I , I , I , I ) H
,
H 6
where
E ( X l , X 2 , . . . , x 8)
It is e a s y frame
of order
THEOREM
is of
to v e r i f y
form
that
(2.55) .
the
3.15.
Let
there
exist
a U(m,l)-semi-decompositions
family
(AI,A2,...,AI,In,n).
orthogonal
matrix
of o r d e r
mn with
is a H ( m , l ) - s e m i - d e c o m p o s i t i o n s . of d e f i n i t i o n
L e t us c o n s i d e r
Wallis
there
the e l e m e n t s
exists
an
integer
x i ~m.
Then
Hi,i=1...8
... + X l H 1
matrices
satisfy
3.9.
= G =
G is t h e m a t r i x approach.
= xIU I + x2U 2 +
the m a t r i x
H ( A 1 , A 2 , . . . , A I) Obviously
Then
and a
Let H ( X l , X 2 , . . . , x I)
conditions
semi-
8m.
Williamson
PROOF.
is a 8 - e l e m e n t a l
s e t { H i }8i=I
Only
A 1 x N I + A 2 × H 2 +...+
desired. this
The
theorem
approach
for
A1 x HI
is p r o v e d .
the p r e s e n t
allows
to
the
74
affirm order ly,
for an a r b i t r a r y 2Sq
for all
following THEOREM
Then s,
one
theorem
rices
of o r d e r s
idea
2Sq,
q = 2m + 3
c)
25
• 67,
24
• 167,
28
24
• 233,
213
re e x i s t s
>
. Let q, q > 3
matrix
23
103,
number,
29
• 179,
107
numbers
precise-
natural
, such
number.
that
for any
2Sq.
existence
of H a d a m a r d
mat-
211
191,
127 23
23
' 213,
• 151
23
• 163
23
• 219,
23
• 223,
slightly
Wallis
result.
• 311
a theorem,
from proof For
strengthening
of t h e o r e m
any n a t u r a l matrix
number
of o r d e r
k
2s~
a n d an o d d n u m b e r
v
the-
, where
(2k - I),
for v ~ 1 ( m o d
4)
K[21og2(V-5)]
(2k - I),
for v ~ 3 ( m o d
4)
-
(The
3.16).
K[21og2(V-3) ] -
for v < 5
Let k = I and
(v+l)/g,
of
s > m
23
k + I
PROOF.
t. M o r e
is a n y
of o r d e r
in p a r t i c u l a r ,
is a p r i m e
an H a d a m a r d
I
matrices
where
b)
3.17.
an a r b i t r a r y
t, t ~ [21og2(q-3) ]
follows
q = 2 TM - I, S > m
is t h a t
with
an H a d a m a r d
a)
THEOREM
S
a number
3.16
let us p r o v e
Now
leading
of H a d a m a r d
is true.
exists
theorem
q the e x i s t e n c e
(Wallis,1976)
find
s > t , there From
s values
3.16.
can
number
v
(v-3)/g
be an o d d n u m b e r .
Then
one
can note
that
with
= f 2,
for v ~ 1 ( m o d
4)
4,
for v ~ 3 ( m o d
4)
g
are
mutually
for n u m b e r s
prime. t
Hence,
satisfying
from
Sylvester
[254]
the c o n d i t i o n
2 t > ((v+1)/g-1) ((v-3)/g-1 =
theorem
=
((v+I)/2-I) ( ( v + I ) / 2 - 3 ) ,
for v ~ 1 ( m o d
4)
((V+I)/4-I) ( ( V + I ) / 4 - 2 ) ,
for v ~ 3 ( m o d
4)
{
}~
follows
that
75
>
and
(V-3)2/4
, for v ~ 1 ( m o d
4)
(v-5)2/8
, for v ~ 3 ( m o d
4)
{
(what is the
same)
21og2(v-3)-2
, for v ~ 1 ( m o d
4), v>3
21og2(v-5)-3
, for v ~ 3 ( m o d
4), v>5
t ~{
there e x i s t
non-negative
integers
2 t = a • v+l g Let us c o n s i d e r Case gonal,
2t-a-b)
+ b
on 4 p a r a m e t e r s
that t h e r e
Xl,X2,X3,X 4
of o r d e r 2 t ÷ l ,
exists
of type
that
an o r t h o -
(2a,2b,2t-a-b,
is
+ (2 t - a - b ) x 3 2
+ (2 t - a - b ) x 4 2 ] I 2 t + 1
a matrix D = F(J,J-2I,
X,Y are
ference
such that
v-3
; suppose
FF T = [2axl 2 ÷ abx12
where
b
g
4), v>3
d e s i g n F(Xl,X2,X3,X4)
Consider
and
2 cases.
I. Let v ~ 1 ( m o d
dependent
a
sets,
incidence that
matrices
is c i r c u l a n t
X,Y) of s u p p l e m e n t a r y
2-{v,~
(-1,+1)-matrices
,~}
satisfying
the c o n d i -
tions XX T + yyT = 2(v + I) I - 2J Then DD T = [2aJ2
+ ab(J_2i)2
= [2t+1(v+1)
+ (2t_a_b) (XX T + yyT) ] x i2t+ I =
- 2a(2t+1)
- 2b(2t-3) ]xi
+ 2t+I
+ [2a(v+1)
SO, D - m a t r i x t ~[2
+ 2b(v-3)
- 2t+I]
is an H a d a m a r d log2(v-3)
- 2]
matrix , v> 3
J4
×
I2t+1
of order
=
2 t+1
2 vt+1
vI2t+1
• v
dif-
for
78
Case other
2. Let
hand,
dependent that
v
3(mod
from Geramita
on
v>5.
Then
theorem
3 parameters
design
2 t+2
[103]
= a(v+l)+b(v-3).
there
exists
On the
an o r t h o g o n a l ,
D ( X l , X 2 , X 3) of type
(a,b,2 t÷2
-a --b) ,
is
DD T = lax12
And
4),
from
the p r o p e r t i e s
v-1 v-3 (v, -~-- , ~
sign
+ bx22
vely,
) satisfies
substitution in
(3.33)
we can
order
that
prove
and
[avJ
Jr'
+ 4bI
x I
an i n c i d e n c e
Jv-2Iv ' B
matrix
for X l , X 2 , x 3
the c o r r e s p o n d e n t
that
in v i e w
EE T = [aj 2 + b(J-2I) 2 + =
sets
(3.33)
I2t+2
B of de-
(v+1)I-J
and by designation
With
- a - b)x32]
the c o n d i t i o n
the m a t r i c e s
as E ( J v , J v - 2 I v , B ) 2t+2v.
(2 t+2
of d i f f e r e n c e
BB T =
By
+
this
design
= [ (2t+2(v+1)
design matrix
of
let us c a l c u l a t e =
2t+2 (2t+2-a-b) (v+l)I
+
orthogonal
is an H a d a m a r d
(2t+2-a-b)BBT]xi
+ b(v-4)J
respecti-
- a(v+1)
- b(v-3)I
+
(2t+2-a-b)j] (a(v+1)
x
+ b(v-3)-
2t+2 - 2t)j] x I
= 2t+2vI 2 t+2
Hence,
E matrix
2t+2v
is an H a d a m a r d
matrix
of o r d e r
2t+2v
with
v>5,
t>
>[21og2(v-5)-3]. Further, der
2tv, So,
rem
where
can
v<5,
show that
there
exists
an H a d a m a r d
matrix
of o r -
t>2.
from combination
of two c a s e s
we get
the
statement
of the
theo-
for k = I. For
the
one
any
theorem NOTE
arbitrary
k > I one
can prove
the
theorem
by a n a l o g y
with
3.12.
3.7.
For k=1
the t h e o r e m
3.17
coincides
with
Wallis
theorem
(1976). THEOREM
3.18.
For
arbitrary
integers
integers
vi,
vi>3 , i = 1 , 2 , . . . , k
77
there exists
an H a d a m a r d
matrix
k s < ~ [21og 2 i=I
COROLLARY
3.14.
of order
(v i - 3)]
Then
there
1=I,3, exists
2 s> 7 [21og2(v i - I - 2)] i=I
From this c o r o l l a r y orders
of H a d a m a r d
k = {103"167,
we have
matrices.
103-233,
i=I,2
Now we will 103 359,
and let kl,k 2 be ar-
an H a d a m a r d
[2(k I + k 2)
in p a r t i c u l a r ,
103 239,
where
- (k - I)
Let v.-=l(mod 4), l
b i t r a r y n a t u r a l numbers. kI k2 2s vI v2 , where
2svl,v2,...,k,
matrix
- I]
the account
introduce 103-419,
of order
set of new
some of them. 127.167,
6 2 .k
127.233,
127'239,
127.359
127.419,
151.167,
151-233,
151,239,
151359,
151.419,
163
163.233
163.239,
163-359,
163~419,
213
213~233,
213'239,
213359,
219.167
219233,
219
219 359,
219419,
223167,
223233,
223239,
223.359
223419,
267167,
267-233,
267239,
267359,
267.419,
283-167,
283 233
283 ~39,
283 359,
283 419}.
167,
27-k,
k 6 {67"103
67 249, 211.k,
67 267, k 6 {107
67
127,
67. 283, 103,
107
67
167
239,
151, 167,
127,
107
67 163, 239 239, 163,
167,
67 213, 359 359,
107 213,
67 219,
67 223,
419~419}
107-223,
107283,
167
179,
179233,
179239,
179419,
269167,
269 419,
251
251,
443 443,
335 335,
67 373,
67 67-103,
107'249,107"267,
269"233,
269-239,
67 67.127}
.
78
§ 4. New m e t h o d for H a d a m a r d m a t r i c e s c o n s t r u c t i o n
We will give in this p a r a g r a p h a "block" a p p r o a c h to c o n s t r u c t i o n of H a d a m a r d m a t r i c e s which allows to obtain r e c u r r e n t formulas for construction of a set of n e w classes of Hadamard,
(for example from exis-
tence of H a d a m a r d m a t r i c e s of order m l , m 2 follows that of H a d a m a r d matrices of order m i ( M 2 / 2 ) n , k -Hall, G o e t h a l s - S e i d e l ,
is an a r b i t r a r y natural number).
Wallis-Whiteman,
A-arrays,
Baumert-
n e c e s s a r y and suffi-
cient c o n d i t i o n s for existence of b l o c k - c i r c u l a n t H a d a m a r d m a t r i c e s , t h e constructions
for b l o c k - c i r c u l a n t and p a r a m e t r i c H a d a m a r d matrices.
F r o m a n a l y s i s of known methods for H a d a m a r d m a t r i c e s c o n s t r u c t i o n namely, Williamson,
Baumert-Hall,
Goethals-Seidel,
Paley-Wallis-White-
man m e t h o d s we c o n c l u d e that H a d a m a r d m a t r i c e s consist of a) Williamson,
Yang, g e n e r a l i z e d Williamson,
A-matrices,
family of
W i l l i a m s o n m a t r i c e s which gather t o g e t h e r a c c o r d i n g to definite rule orthogonal) b)
Some
designs
(Baumert-Hall,
(as a
G o e t h a l s - S e i d e l arrays et.al) .
"base" array with definite p r o p e r t i e s
s u p p l e m e n t e d with a
number of new rows and columns for r e d u c t i o n to a H a d a m a r d matrix. The p r o b l e m arose: matrices
(namely,
to synthesize an H a d a m a r d m a t r i x from the same
from complete and incomplete matrices).
The p o s s i b i l i t y of such a c o n s t r u c t i o n was d e m o n s t r a t e d by Sylvester
(1967)
as e a r l y as last century by c o n s t r u c t i o n of of H a d a m a r d
m a t r i c e s of order 2 n. The c o n s t r u c t i o n for this a p p r o a c h is very simple and has the following form:
H(k+1)
=
H(k)
H (k)
-H (k)
(4.0)
I
I where H I ) 6 {_+D1, +D2, +D3,
°
H(k)
+_D4}
,
(4.1)
79
N o w let us c o n s i d e r PROBLEM incomplete must
4.1.
the m a t h e m a t i c a l
Let Qi,j'
Hadamard
i=1,2,...,m,
matrices
impose on the m a t r i c e s
H
an H a d a m a r d
formulation
of o r d e r Qi,j
of the q u e s t i o n s .
j=1,2,...,n, (p,q), pm=qn.
be c o m p l e t e
and
What conditions
one
to b e c o m e
QI,1
QI,2
"'"
Q1,n
Q2,1
Q2,2
"'"
Q2,n
Qm,1
Qm,2
"'"
Q m,n
pm
m a t r i x of o r d e r pm.
Let us give a l s o two investigated PROBLEM
special
c a s e s of this p r o b l e m
w h i c h will be
in this work. 4.2. L e t Qi'
m. W h a t c o n d i t i o n s
one m u s t
Q0
H
i=I,2,...,n-I
be H a d a m a r d
matrices
of o r d e r
impose on Qi to b e c o m e
QI
Q2
"'"
Qn-1
Qn-IQ0
QI
"'"
Qn-2
mn . , ,
, .
(4.2) QI
Q2
Q3
"'"
Q0
2 = I n x Q 0 + Un x Q I + Un
an H a d a m a r d
p2 ) and numbers
be
x Qn-1
m a t r i x of o r d e r mn.
PROBLEM ...,k
n-1 x Q 2 + ... + Un
4.3.
Let
(-I,+I)
ei 6 { 0 , - I , + I }
s q u a r e or r e c t a n g u l a r
(ql,q2) , r e s p e c t i v e l y . ~i
(n+1) 3
k = X ~.X.x i= I 1 i
be
(0,-I,+I)
matrices
What conditions
and on t h e s e m a t r i c e s
H
, Xi
H.(n) 3
j=1,2,...,k
to b e c o m e
and H i , i = 1 , 2 , . . .
of d i m e n s i o n s
one m u s t
impose
(PI' on the
the square m a t r i x
,
(4.3)
80
an
Hadamard
matrix
Note
that
of
problem
Note
also
case
(Sylvester on
(4.0)
for
with
the
limitations
the
that
be
first
method
method)
was
of
in
form
rewritten
4.2
is a
special
for
Hadamard
the
= X × H(n)
same
(see
matrices
approach:
construction in
fact
the
relati-
4.3)
+ y x H(n)
,
matrices
X
1 0 [
=
,
Y
=
I0 1
0 -I
and
problem
4.3.
H(n+1) where
n
definite
(1867) can
any
H(n)
is
an
Hadamard
This
paragraph
is
I
matrix
obtained
denoted
to
at
0
the
n th
investigation
of
step. the
problems
4.2
and
4.3. LEMMA ting
of
4.1.
Let
matrices
H
be
mn
a block-circulant
(blocks)
Qi'
, i,j
a)
Q i Q TJ
= Q j Q T~
b)
Qn-i
= Qi'
i=0,1,...,n-1
matrix of
of
order
order
m
mn
consis-
. Let
= 0,1 ..... n-1
(4.4)
or
are
(4.5)
i=1,2,...,n-I
true. Then
Hmn
HT mn
= I
n
× F 0 + U×
F1 +
" ""
+ un-1 ×
Fn- 1
(4.6)
'
with n-1
Fk =
k-1
IE Q i Q - k
+
i=k for
k=0
second
THEOREM an
odd
number
item
4.1.
Let
T
I QiQi+n_k
, k=0,1,...,n-1
(4.7)
i=0 is a s s u m e d H
consisting
mn
to b e
zero.
be
a block-circulant
matrix
of
Hadamard
Qi'
matrices
of
order
i=0,1,...,n-1,
mn(n
of
is
81
order be
m.
an
Suppose
Hadamard
also
that
matrix,
it
(4.4)
or
(4.5)
is n e c e s s a r y
are
and
true.
Then
sufficient
the
for
H
Kin
validity
to of
relation
n-1
NECESSITY. mn
k-1
T QiQi_k
X i=k
Let
Q
X i=0
+
H
T iQi+n_k
nun b e
= 0
(4.8)
, k=1,2,...,n-1
a block-circulant
Hadamard
matrix
of
order
namely,
Hmn
Then
we
have
on
the
one
n-1 X i=0
=
hand
H m n H mT n
from n-1 Z k=0
=
U i x Qi
(4.9)
Lemma
Uk x Fk
4.1.
,
(4.10)
with n-1 Fk
and
on
=
X i=k
the
other
H
mn
Now
is
from
an
+
mn
HT
mn
comparison
from
the
structure
mnI
=
Hadamard
I x F0
but
T QiQi+n_ k
x i=0
(4.10)
matrix
from
(4.12)
we
.
of
order
with
+ U x FI +
of
(4.11)
mn
matrix
(U k x F k)
Hence,
, k=0,1,2,...,n-1
hand
H
since
k-1
T QiQi_k
...
H
mn
(Uq×Fq)
nun.
(4.11)
we
+ U n-1 x F n _ I = m n I m n
we
= 0
have
, k
obtain
uP x F
= 0, P
obtain
p=1,2,...,n-1
~ q
(4.12)
82
So,
on
F
= 0, p = 1 , 2 , . . . , n - 1 P S u f f i c i e n c y is r e a d i l y
F 0 = mI m
which
i=0,I,2,...,n-I
the n e c e s s i t y obtained
follows
are
from
Hadamard
is p r o v e d .
from Lemma
(4.7)
and
matrices
4.1
and
from
the
f r o m the c o n d i t i o n s
of o r d e r
m. The
theorem
relatithat
Qi'
is p r o -
ved. L e t us n o w c o n s i d e r ruction matrix
necessary
of b l o c k - c i r c u l a n t
and
Hadamard
sufficient
matrices
conditions
with
for c o n s t -
the e l e m e n t s
of Qi-
only.
Let
us d e n o t e
,xl
Qk = H(xlk) ,X~ k) ,...,Xl(k)),
,...,X
THEOREM Hadamard ...m-1
4.2.
matrix
) is a A - m a t r i x , For
(BCBSH)
generated
...X I) of o r d e r following
by k
of o r d e r
some
~{-I,+I}
k=0,1,...,m-1.
of b l o c k - c i r c u l a n t ,
mk
dependent
which
consist
on p a r a m e t e r s
it is n e c e s s a r y
where
and
block-symmetric
of b l o c k s A-matrix
sufficient
the
Qp,p=0,1,2.
H(Xl,X2,...
validity
of the
conditions:
I. X i(p)
2.
the e x i s t e n c e
Xl!k)6
= X (p)m_i ' i = I , 2 , . . . , m - I ,
Ix!P) I : I , i=1,2 . . . . m-l, 1 "
p=I,2,...,i
(4.14)
p=1,2 . . . . 1 "
(4.15)
3. a) I i~j,i~m-j
1 ( X q=1
x ! q ) x ! q)) l 3
i+j=p
for the
odd
or
= -i
i+j=m+p
(4.16)
m 1
b)
z i+j=2p
or
{ z i+j=m+2p
for the e v e n NECESSITY. matrices
or
1
= 2k
3
i+j=m+2p+1
1 ( I q=1
is a B C B S H
matrix
Z i+j=2p+1
x!q)x! q))
q=1
x }qq)))x'!' l 3
= 0
m. L e t Hmk
(blocks)
generated
by
some
of o r d e r
dependent
of
mk w h i c h
consist
1 parameters
of
A-mat-
83
rix
of o r d e r
according
the t h e o r e m
to b l o c k - s y m m e t r y
Further, rix,
k. F r o m
since
Qi'
4.1
we h a v e
we h a v e
the
i=0,1,2,...,m-1
(4.8)
relation are
and F 0 = mI m
Qi
= Qm-i
obtained
from
and
too. some
A-mat-
then
--~
+ x2
...tA 1
A1
) I k,
i # j
(4.17)
is true. The r e l a t i o n
can be r e w r i t t e n
2k X 1 i+3= p
(~ (i). (j) ~I AI
in f o r m
. (i). (j)+. + (i)x~J)) + x2 ~2 "" Xl
or
Ik
=
0, (4.18)
p=1,2,...,m-1
i+j=m+p
L e t us c o n s i d e r
2 cases:
1) m
is an o d d n u m b e r ;
2) m
is an e v e n
In the
first
number,
case
for e a c h
p
2 2 X I + X 2 +...+ takes
part.
to the
This
item
is 1
(since
(4.16)
one
i t e m of
form
2 X1
(4.19)
IXil
= I, i = 1 , 2 , . . . , i )
it e n t e r s
in-
sum ,.(i). (j) Z ~AI AI i+j=p
when
in
p
is an o d d n u m b e r
(x~i)xl j)
. (i). (j) + x2 A2
and
into
the
÷'''+
..(i)~(j) A1 ~i ) ,
(4.20)
sum
(i)x~ j) + X2 +'''+
. (i). (j)) A1 A1
(4.21)
i+j=p+m when
is
p
is an e v e n
number.
Hence,
for e v e r y
1
i=j
for
rewritten
or
in f o r m
p
i+j=m;
the
relation
that
means
(4.16) that
includes
the e q u a l i t y
the
item which
(4.16)
can be
84
(x~i)xl j) + x J i ) x ~ j) + • . . +
E
i+j=p
or
x l(i)x~j))
= -i
(4.22)
i+j=m+p
i@j, i~m-j
The
second
part
is an H a d a m a r d
proved
follows
the n e c e s s i t y
let us c o n s i d e r
consider
theorem
f r o m the
condition
that
H
matrix.
So we h a v e Now
of the
the
for o d d
second
case
m.
when
is an e v e n
m
number.
We
2 cases:
a) p = 2q (4.23) b)
p = 2q+I,
L e t p = 2q
q=I,2,...
, then
(x~q))2
Hence,
+
(4.16)
(x~q))2
for p = 2q
let p = 2 q + 1 .
des
the
items
i=m-j
this
Hence,
to
part
So,
a) Let
{x~i)}m-1
i=0
the
(for the v e n Rewrite
(4.16)
E or i + j = m + 2 q + 1
SUFFICIENCY.
satisfying
sum of
there
'
(4.22)
which
are
= 1
(4.24)
in f o r m
+. . .+ x ~ i ) x ~ J ) )
first
part
on the o t h e r
the
items
with
even
exist
of the
= -21
(4.16) for
inclu-
i=j,
sum of
indexes.
i.e.
we o b t a i n
i=0
of hand
(x~i)x~ j) + x ~ i ) x J J ) + . . +•
{X 2~(i)}m-1
the p r o o f
the
indexes,
includes
a n d b)
conditions m
hand,
we c a n o d d a l s o
by c o m b i n a t i o n
(x~q))2
+ X 2(i)x~j)
on the one
odd
of
(4.16)
i+j=2q+l
(xli)xl j)
Then
with
+...+
2 items
we h a v e
X i~j,i~m-j Now
includes
the n e c e s s i t y
x I(i)x}J))
: 0
for e v e n
p too.
sequences
'
..
"'
{x~i) }m-1
theorem.
is a n a l o g o u s ) .
i=0
Let
m
be an odd n u m b e r
85
2kl'(i)" (J) " (i)v(J)+..+ E -~'~I AI + A2 ~2 " i+j=p or i+j=m+p Further,
substituting
both sides by 2k/l
for l,(X~i)) 2 +...+(x~i)) 2
" (i)x~J))Ik-i Xl
= 0
and m u l t i p l y i n g
we obtain -~1~12k " (ilx j) + x2(i)x j)+'''+ ~i
AI
(4.25)
)Ik = 0
i+j=p or i+j=m+p NOW let us compose ding to c o n s t r u c t i o n ...,X 1
the matrix
of
sequence
A~matrices
{-~i}m-li=0 of order
substituting
k
the variables
accorXI,X2,
for the numbers
xI
c {xli) }i=0 m-1
(i)}m-1 i=0
, X 2 6 iX 2
'''"
,
Xl £
{x~i)}m-1 i=0
respectively. The matrices lity Qi = Qm-i
Qi'
i=0,1,...,m-1
' since X i = Xm_i,
ined from A - m a t r i c e s
(4.25)
constructed
will satisfy the equa-
i=I,2,...,m-I.
can be rewritten
So, since Qi are obta-
in form
T l QiQj = 0 i+j=p or i+j=m+p
or p m-1 T EQi QT + E QiQp+m_i i=0 P i=p+l That is the condition
= 0
(4.8) of t h e o r e m 4.1 holds.
So we obtain that
matrix H =
is the b l o c k - c i r c u l a n t follows
m-1 I Ui×Qi i=0
Hadamard matrix.
from equality
Qi = Qm-i
' i=I,2,.°.,m-I
The block-symmetric
condition
86
The sufficiency COROLLARY
is proved.
4.1. Let
Ixi(J) I = 1, i=I,2,...,1,
j=1,2,...,m-1
I) x1(0)x1(J)
+ x2(0)x2 (j) +...+ Xl(0)Xl (j) = -1/2,
l=2p
2) x1(i)x1(J)
+ x2(i)x2 (j) +...+ xl(i)Xl (j) = 0, i~j,
(4.26) i=I,2 .... , m-1 Let also there exists dependent ...,X I) of order
on
1
parameters
mk.
Now let us try to find the conditions nimal number of by A - m a t r i c e s
A-matrix Q(XI,X2,
(different)
matrices
which are necessary
and b l o c k - c i r c u l a n t
allowing
from the set of matrices
for c o n s t r u c t i o n
block-symmetric
to pick out the migenerated
of b l o c k - c i r c u l a n t
Hadamard matrices
(BCH and BCBSH
respectively). Let Qi'
i=0,I,2,...,m-I,
be Hadamard matrices
which one must pick out BCH or BCBSH.
Suppose
of order
k
from
that Qi were obtained
from an array of type A[4,k,Ik]. Following
statements
STATEMENT
4.1. Let
are true.
(4.27)
Hmk = I x Q0 + u × QI +''" + u m-1 x Qm-1
where
m
is an odd number and Qi = Qm-i'
first row Q0 includes kql/p, of remaining blocks Qi'
qi=2,4
i=1,2,...,m-I
positive
i=I,2,...,m-I
elements.
. Suppose that Then first rows
include kq2/p , q2=I,3
positive
elements. STATEMENT i=I,2,...,m-I q2=I,4
4.2. Let in . Suppose
NOTE 4.1. Wiiliamson
m
Then first rows of remaining blocks Qi'
, include kq/p, q=2,4 If matrices
array,
be an odd number and Qi=Qm_i ,
that first row of matrix Q0 includes kq2/p,
positive elements.
i=1,2,...,m-I
(4.27)
Qi'
positive
i=0,I,2,...,15
elements. are g e n e r a t e d
from a
that is are of form:
Q0 = W(-I,-I,-I,-I) , QI = W(-I,-I,-I,-I) , Q2 = W(-I,-I,
I,-I)
,
87
Q3
= W(-I,-I,
I, I)
' Q4 = W(-1,
I,-1,-1)
' Q5 = W(-1,
Q6
= W(-I,
I,-1)
' Q7 = W(-1,
I,
' Q8
Q9
= W(
I,
I,-I,-I,
I)
Q12 = W(
I,
I,-I,-I)
Q15 = W(
I,
1,
then
from
first
Qi'
I,-I,
, Q13 = W(
I,
1)
I,-I)
I,-I,
I)
I)
,
= W(1,-I,-I,-I)
,
, Q11 = W(
I,-I,
I,
I)
,
, Q14 = W(
I,
I,-I)
,
I,
I)
statements
from blocks Hadamard
I,
, QI0 = W(
I,
I,-I,
4.1
a n d 4.2
i=0,I,...,15,
matrices,
one
follows
one m u s t
of w h i c h
that
have
appears
for c o n s t r u c t i o n
only
only
9 different
once
and
BCBSH types
in f i r s t
of
place
of
row-block.
N o w the
question
for a g i v e n Let number
4m
to f i n i s h
us note of e a c h
BCBSH
by
first
rows
The
type
j-th
4.2.
Let
m
and
4.2.
one m u s t
fix
in
(4.27)
Hmk u n i q u e l y . to d e f i n e
for a g i v e n
4m
the
Qi E { Q 0 , Q 1 , . . . , Q 1 5 } = Q " a s s e m b l a g e "
in m a t r i c e s notes
matrices
allowing
Denote
by
#i,~2,#3,#4
QiEQ
the
corresponding
statements
be any n a t u r a l
are
Representation
(4.29)
of
sum of e l e m e n t s to f i r s t
of
r o w of BCNSH.
true.
number.
Then
4 = ~ #i(m-#~i) i=I
m(m-1)
NOTE
defining
matrices
(4.27).
following
LEMMA
how many
an a p p r o a c h
design of
arose:
(4.29)
is n o t
unique.
For e x a m p l e
for
m=25 I ~I
I = 13,
2 ~I
~2
2 = ~2
NOTE
4.3.
element
in the
then
2
2 = ~4
~4
~i = 1,2,3,4 first
sign,
(4.30) (4.27)
depends
place
then
= 9
= 15
If r e p r e s e n t a t i o n
of
is "+"
I = 15,
= ~3
eveness
place
I = ~3
of the
is an H a d a m a r d
matrix,
then
on the
sign
of c o r r e s p o n d i n g
matrix
first
row.
Moreover
if in
i-th
is "-"
sign,
~i is o d d a n d
if in i-th p l a c e
~i is even. STATEMENT
4.3.
Let
the m a t r i x
F0=Q15
in r e p r e s e n t a t i o n
(4.27)
be
88
fixed
and
suppose
to c o n s t r u c t system
that
the
from
row-block
every
Fi,
of B C B S H .
i=1,1,...,8 Then
one
needs
X. s a t i s f i e s l
the
X i pieces following
of e q u a t i o n s :
X I + X 2 + X 3 + X 4 = m - ~1 X I + X 2 + X 5 + x 6 = m - ~2 X I + X 3 + X 5 + X 7 = m - ~3 X 2 + X 3 + X 5 + X 8 = m - ~4
for
NOTE
4.4.
which
Q0
Xl
+ X4
+ X6
+ X7
= ~4 -
X2
+ X4
+ X6
+ X8
= ~3
- I
X3
+ X4
+ X7
+ X8
= ~2
- I
X3
+ X6
+ X7
+ X8
= ~I
- 1
There
that
means
cient
for
uniquely
4.37.
using
It
an e x a m p l e
in r e p r e s e n t a t i o n
rent,
Now
exists
that
fixing
definition
, ~2
= ~3
X I = X2 = X3 = 8 From
here
and
H 4 . 3 7 = I x Q0
from
+ U × Q3
+ u 14 xQ1
+ U27xQ1 x Q8
and
of
other
block-matrices
let
= ~4
= 17
blocks us
give
of the
order
Qi a r e is n o t
4.25
diffesuffi-
BCH. BCBSH
of o r d e r
,
, X4 = X5 = 0 of
+ U3xQ7
+ u16xQ2
+ u11xQ2
we h a v e
+ U5xQ6
xQ8
+ U6xQ3
+ U12xQ3
+ u 18 ×Q7
+ U24
+ U30×Q6
+ U34 xQ 7 + U 3 5 x Q 3 + U 3 6 x Q 3
(4.31)
+ U4 ×Q8
+ U17×Q1
U22×Q2 + U23xQ1 + U29xQ2
, X6 = X7 = X8 = 4
system
+ u 9 xQ I + u 1 0 x Q 1
+ U28xQ1
matrix
that
+ U2 xQ3
+ U 1 5×Q2
+ U21 xQ 2 +
coincide
remaining
result
solution
+ U7xQ 6 + U8×Q2
x QI
of
to c a l c u l a t e
~LI = 13
Hadamard
of d i a g o n a l
above-mentioned
is e a s y
I
of
(4.27)
(4.31)
+ U31xQ3
+ u I 3xQ8
+ U I9xQ7
+ U25xQ3
+
+ u 20
+ U26xQ2
+ U32×Q6
+ ×
+
+ u33x (4.32)
89 So, a c c o r d i n g
to the
statements
the m a t r i x F 0 = Q15 = H needs
is fixed,
( or f e w e r m a t r i c e s
deration,
t h e n one n e e d s
different
type n a m e l y
HI = Q15
' H2 = Q2
statement
4.1 and 4.2 we h a v e t h a t if in
Further,
the
matrices
for c o n s t r u c t i o n
t h e n for c o n s t r u c t i o n
f r o m Q. If the s i g n for c o n s t r u c t i o n
' H4 = Q4
4.3 a l l o w s
(4.27)
B C B S H one
is n o t t a k e n
into c o n s i -
B C B S H 4 or fewer m a t r i c e s
of
' H5 = Q8 to d e f i n e
of BCBSH.
the n u m b e r
The p r o b l e m
of e a c h type
of t h e i r
location
re-
m a i n s open. Let us i n v e s t i g a t e
this problem.
Rewrite
(4.27)
in the f o l l o w i n g
form: H = H I × I + X 2 x H 2 +...+ X 5 x H 5 Using
the
r i c e s Xi,
structure
of m a t r i c e s
i=I,2,3,4,5
Hi,
4.33)
i=I,2,3,4,5
are c i r c u l a n t
and
and supposing
symmetric,
that
that mat-
is XiX j =
= X j X i, X i = X Ti ' Xl = I k , we o b t a i n 2(X
T IX2XQ2
T T T + X2XI×Q2 + XlX3XQ4
T x T + X~X~xQ~ + T T + X3Xl Q4 i ~ , X4XI×QI +
+ X l X 57XTQ
+ X 5 X lTX Q 7T ) + 4 ( X 2 X ~ x K 3 - X 3 X 23×TK
T + X2X5xK2
T - X5X2×K2
+ X4X5xK 3 - X4X5xK3 + (X3xQ4
÷ 4t(x2x
~ + X3×Q4)
x3x
÷ ,x4x 5 + 4 Z XiXixl4 i=I
- X 4 X 2Tx K 4
T T T T + X4X3xK 2 - X3X4xK2 + X3X5xK4 - X5X3xK 4 + 5 + Z X i X x HiH = 2 [ ( X 2 x Q 2 + x2x Q ] + i=1
+ (xT4 xQ]
K3÷ (x2x ,x3x
=
+ X 2 X 44xTK
÷ x 4 ~T, ~1~
÷ cx~xQ7 ÷ xsxQ~) ~ ÷
÷ (x2x ,x4x
+ x x ,x 31 ÷
2[X2x(Q 2 + Q ) + X3x(Q 4 + Q ) + X4×IQ I + Q ) + 5 + x5x (Q7 + O~) ] + 4 Z x i X ~ x I 4 = - 4 ( X 2 + X3 + X 4 + x 5) xI 4 + i=1 5 5 5 + 4i=]XiXixI4X T = 4(X Xi + Z XiX ~ + I k) xI 4 . i=I i=1
+
90
Further,
HH T
supposing
that
5 (- Z X i=I i
= 4
HH T
= 4kI4k
5 Z XiX ~ i=I
+
+
, we
I k) × 14
have
= 4kI4k
SO,
5 Z Xi(X i i=I
Hence,
it
4.5.
XI,X2,X3,X4,X
Let
5 of
* X~ 3
= 0,
i ~
2.
X I + X2
+ X3
+ X4
3.
5 I X (X i i=I i
the
matrix
Ik)
H
that
we
will
X I =
I9,
X2
I11 , X 2
= U5
also
the
(0,-1,+1)-matri-
conditions:
..... 5 of
order
k.
Hadamard
matrix
of
Qi
leads
to
statement
4.5.
I)I k
is
the
matrices the
examples
+ U 6, + U 8,
of
such
X3 = U2 x3
= u2
+ U 7,
= U
- U2
- U9
+ U I0,
= U
-
+ U5 - U6
construction
matrices
+ U 5,
X4 X4
of
= U3 = U3
order
4k. of
5
orders
+ U 4, + U 6,
X3
= U3
+ U 8,
- U7
+ U8
- U9
X5 x5
X4
= 0; = U4
= U7
+ U5; + U 7,
+ U6;
I11 , X 2
X3 = U2
is
= U
symmetric
(-1,+1)-matrix
of
= U
k=9,
X I =
(k -
some
, X2
X5
=
circulant
i,j=1,2
+ X5
give
17
X I =
j,
satisfying
X I =
k=13,
I)I k
satisfying
5 X H × X i=I l l
=
k=7,
k=11,
k
"numeration"
(0,-1,+1)-matrices Now
(k -
exist
order
X. ±
Note
It
there
I.
Then
=
holds
STATEMENT ces
I k)
+ U3
U4
+ U I0
+ U 11
t
X4
exists
a
= X5
+ U 12,
= 0
true.
STATEMENT
4.6.
H4m
=
Let
there
m-1 X
Ui
i= 0
F
(a,b,c 1
BCH
d) '
of
order
4m.
Then
(4
34) °
91
with
Fi
= Fi(a'b'c'd)
A
* A,
b
c
d
a
d
c
d
a
b
c
b
a
i=0,1,...,m-1
=
is a b ! o c k - c i r c u l a n t NOTE
= Qi
4.5.
The
parametric parametric
Hadamard
matrix
matrix
(4.34)
of
order
constructed
4m. is n o t
an
A-mat-
rix. THEOREM lant
of
4.3.
order
S
mn
Let
n.
A (i) P,q
For
be
m
' P'q
=
1,2,...,m,
is
a circu-
A11
A12
•..
Aim
A21
A22
•..
A2m (4.35)
=
represented
order
l•a p(i) , q ~ % n-1 i=0
matrix
Am I
to
=
it
as
Am 2
=
Amm
a block-circulant
is n e c e s s a r y
Hmn
"""
I × Q0
and
matrix
sufficient
+ U × QI
+
"'"
+ Un-1
the
×
consisting validity
Qn-1
of
of the
blocks
of
relation
'
where
021i) Qi
"'"
Ulm
"'"
U2m , i=0,1,...,n-1
= .o
Q(i) ml
•
_(i) ~ m 2 "'"
.
~(i) Umm
(4.36)
92
NOTE
4.6 • If
ric m a t r i c e s ,
(i) ~ n-1 {Ap,ql i=0
then
there
4.7.
If t h e r e
then
there
' P'q
= 0,I,...,m-I
true
representation
exists
a Williamson
are
are
circulant
(4.36)
and
symmet-
Qi=Qn_1 ,
i=I,2,...,n-I STATEMENT of o r d e r
4m,
generated
by a W i l l i a m s o n
mard
matrices STATEMENT
where
PI'
rices, damerd
of o r d e r 4.8.
are
a~,~2,...,a n
are
array
Hadamard
4m
matrix
constructed
matrices,
that
from
is f r o m
Hada-
exists
a BCBSH
in p a r t i c u l a r arbitrary
of o r d e r double
p 0 . p ~ 1"P2C~2
orders
integers,
P0
o
I
Pn n ,
of W i l l i a m s o n
is the o r d e r
mat-
of an Ha-
m a t r ix. 4.2.
If t h e r e
of o r d e r
mn,
generated
by B a u m e r t - H a l l
COROLLARY Wi!liamson
Then
a BCH of o r d e r
Hadamard
4.
There
P2'''''Pn
COROLLARY
design
exists
type
then
4.3.
there
Let
matrices,
matrices)
exists of o r d e r
exists array
a BCH of o r d e r
exists
of o r d e r
m
generated
mn
of type
a type
Hadamard
constructed
matrix from
( S l , S 2 , . . . , S l , I n)
( S l , S 2 , . . . , s I) o r t h o g o n a l
consisting mn
type
matrices.
be a f a m i l y
a B C H of o r d e r m
a Baumert-Hall
Hadamard
{Ai}l=1
let t h e r e
O(Xl,X2,...,Xl)
there
exists
of e l e m e n t s
consisting
_+Xi, Xi~0.
of b l o c k s
by the o r t h o g o n a l
design
(Hadamard 0 ( x 1,x2,x 3,
.... x l) • Using with
the
all a b o v e - m e n t i o n e d family
STATEMENT Now
let us
we o b t a i n
of W i l l i a m s o n
4.9.
There
investigate
an i n t e r e s t i n g
fact
connected
matrices.
exists
no type
the p r o b l e m
4.3.
(1,3,I 3) W i l l i a m s o n Find
the n e c e s s a r y
family. conditions
for
H(n
to be an H a d a m a r d
H(n
+ I) =
matrix,
k E X i x Hi(n) i=I that
+ I) H T ( n
(4.37)
is
+ I) =
(N + 1)In+ I
(4.38)
93
From
(4.29)
H(n
we h a v e
+ I)HT(n
k k Y X,X T x H , H T + i=1 i i i i i=I
+ I) =
k j=1
i 3
i 3
i~j
=
Let
us c o n s i d e r a) X i * xj b)
(n +
two g e n e r a l
1)In+ I
cases:
~ 0,
I~j,
i , j = 1 , 2 ..... k
X.z * X T] = 0,
i~j,
i , j = 1 , 2 ..... k
HT = I × T T nun Q0 + U × QI +'''+
that
(4.39)
is H T mn
is a B C B S H
It is n a t u r a l
U n- I ×
of o r d e r
to a s s u m e
(4.40)
QT n-1
(4.41)
nun
in b o t h
cases
that
k Z X i=I l is a s q u a r e
or r e c t a n g u l a r
L e t us d i v i d e I. Let then
(-1,+1)-matrix.
the case
H I = H 2 =. ..=H k
the n e c e s s a r y
(4.42)
a)
into
be H a d a m a r d
conditions
several matrices
of v a l i d i t y
items. of o r d e r
of r e l a t i o n
(PI'P2)=PI' (4.38)
k i=I
xixT
k k Z X x.xT i=lj=1 13
Note
that
(4.44)
XiX~
will
be
+ X 3.xTl = 0,
=
(4.43)
PIIpl
= 0,
i ~ j
satisfied
i~j,
(4.44)
provided
i,j=1,2,...,k
(4.45)
94
-matrices form a
Xi,
k-elemental
above
satisfying
frame;
conditions
theorem
are
4.4.
for m a t r i x
(4.37)
satisfying
the c o n d i t i o n s
2. N o w
let matrices
the n e c e s s a r y
x 1
NOTE
4.9.
The
xT ]
and
Hi,
H2 = HT 2 The
= 0,
(4.43)
and
(4.45)
i.e.
the
following
condition
=
x ]
square
xT J.
, HI
matrices
,
4.3.
Hadamard
(4.46)
For e v e r y matrices
and
become
H2 ,
satisfy
m.
for eve-
i=1,2,...,k,
(4.45). the c o n d i t i o n
(4.46)
Xi,
of
(4.40)
is
(4.47
i=1,2,...,k
(4.47)
form a
of o r d e r
n
satisfying
k-elemental
the c o n d i -
hyperframe.
form a hyperframe,
for
for k=2
Hadamard
Hi,
L e t us d e f i n e
al,a2,...,a m
(4.43),
Xi,
n+1 mPl
i,j:1,2,...,k
i=I,2
of m a t r i x T H 2 = [a2,
= 0
matrix
i=I,2,3,4,
H . H T + H . H T = 0, 1 j 31
PROOF.
of o r d e r
of o r d e r
= -H TI ;
condition
LEMMA
matrix
of m a t r i c e s
of v a l i d i t y
matrices
Xi,
Hadamard
i 9 j
HIH T 2 + H2H~
ruct
too,
matrix
(4.42),
i=1,2,...,k
(4.40) , (4.42) , (4.43) square
are
sufficient
(4.40),
k k Z Z H.HT i=]j=1 13 Then
Hk
to b e an H a d a m a r d
it is n e c e s s a r y
The
conditions
Let H I = H 2 =...=
ry n,n ~ 2
tions
(4.42),
T H2 = H2 ,
for k=2,
sufficient
(4.40),
is true.
THEOREM Then
i=1,2,...,k
H I of o r d e r
of o r d e r
m
m
one
can const-
, such that
i / j, i , j = 1 , 2 , 3 , 4
an operator
Ei,
i=2,3,4,
(4.48)
converting
H I to m a t r i c e s
-a I, a 4,
-a3,...,
a m , -am_1 ]T
;
the
rows
95
H3 ,
H T3 =
H4
H T4 = [a 4 , a3 , -a2 , - a 1 ' ' ' ' '
t
[a 3 , -a4 , -al , a2 , ... , a m _ I, -a m , - a m _ 3,
a m , am_1~
- a m _ 2,
a m _ 2 IT
- a m _ 3 ]T
respectively. It
is e a s y
satisfy
the
condition
Combining THEOREM satisfy
the
above
matrices
together
with
matrix
H
I
(4.48).
Let
Hadamard
condition
(4.41),
hyperframe
we obtain matrices let
of o r d e r
the p.
Hi,
i=I,2,...,i
matrices
Then
for
of o r d e r
Xl,X2,...,X every
1
natural
m
form
a
number
n
matrix
H(n
is a n
Hadamard
COROLLARY rices p.
that
above-mentioned, 4.5.
1-elemental the
to v e r i f y
Xi,
matrix 4.4.
every
n
The
proof
of
4.5
but
which
we will
H I = H0
tisfying pose
H I be
the
relations
an
exists
corollary
give
another,
for one
=
1=2,4,
there
is n e c e s s a r y
matrix
Let
the
I)
of o r d e r
i=I,2,...,i,
Then
+
the
can
conditions
1 X X. x H, (n) i=I i 1 mp
n+1
Hadamard form
a
an Hadamard 4.4.
matrix
follows
constructive
construct
from proof
of order lemma of
matrices
matrices
m
, let mat-
hyperframe
According
Hadamard
Using
of order
1-elemental
applications.
(4.48).
matrix
to
mp
4.3
this
of
and
theorem
corollary
lemma
4.3
of o r d e r
constructed
order
n+1
let
for m
sa-
us
com
for
1 = 2 H i + l , I = X I x Hi, 1 + X 2 × Hi, 2
,
H i + l , 2 = - X 1 x Hi, 2 + X 2 x Hi, I , i = 4 H i + l , I = X I x Hi, I + X 2 x Hi, 2 + X 3 x Hi, 3 + X 4 x Hi, 4
,
H i + l , 2 = Xl x Hi, 2 - X2 x Hi, I + X3 x Hi, 4 - X4 × Hi, 3
,
96
H i + l , 3 = X I x Hi, 3 - X 2 x Hi, 4 - X 3 x Hi, I + X 4 x Hi, 2 , H i + l , 4 = X I x Hi, 4 + X 2 x Hi, 3 - X 3 x Hi, 2 - X 4 x Hi, 1 .
COROLLARY
exists where
4.5.
For every
an Hadamard Pi
matrix
natural
el,
U a. i Pi I '
of order
is in p a r t i c u l a r
numbers
Pi
there
E { 1,2,...,49,50},
2 of e x i s t i n g
the order
az,...,~ n
Williamson
type
matri-
ces. COROLLARY and Hall
4.6.
2-elemental array
q~1(mod
re e x i s t s Note nite
(1898)
4)
is the p o w e r matrix
exists
m
are
square
array
there
4.7
an Hadamard
4m(q+l) k
4.5 and
4.6 a l l o w
matrices
and Baumert-Hall
includes
for
by means
k=1.
of n u m b e r - t h e
matrices.
4 i=lXX i X i = p i p
number.
of o r d e r
§. L e t us c o n s i d e r the p r o b l e m 4.3 X. l
. Then
of a prime
the corollaries
corollary
a Baumert-Hall
of o r d e r
Let there
of Hadamard
results
exist
of o r d e r
exists
n
a Baumert-
ran .
an Hadamard
class
and that
4.7.
that
there
hyperframe
of o r d e r
COROLLARY let
Let
Then
Paley
retic for
matrix Then
of order
for every
to c o n s t r u c t arrays (1933)
an
4m k
; the-
infi-
of n e w o r d e r s and
Scorpis
methods. k=1,
Hi(1)6{Di}~= I and
provided
m
the
following THEOREM
theorem
4.6.
For
H(n
, X i * X.3 = 0,
the r e l a t i o n s
4 + I) = X X. i=I l matrix
ry and
that
H. (n) 1
of o r d e r
matrices
2p n
Xi,
(4.50)
(n
= -X
-X T i+z i+1
'
is a r b i t r a r y )
i=I,2,3,4,
ditions
Xi+,X~ +2
(4.49)
i,j:I,2,3,4
is true.
to be a n H a d a m a r d sufficient
i~j,
i=0,2
shoul
it is n e c e s s a satisfy
the c o n -
97
XIX3T + x3xT = x2xT + X4x2T
(4.51)
X2X T3 + X3X T2 = XlX~ _ X4X TI NECESSITY. According to condition
Hi(1)£{D1,D2,D3,D 4}
we can
write 4
Hn+IHT+I = 2 > - x i x T × i=I
I2 + (XIX2 + X2X 1) x I2 +
T × + 2 ( X l X 3 + X3Xl)
II10 °11 2
+ 2(X2X 3 + X3X T2 +
x3x~)×ll_1 oil
+ 2(XlX4T -
0
0
÷ (x3x~ + x4x[Ixll I
0
X4X~) ~11 1
-I 0 I
I
+ 2(X2X ~ + x4xT)x
0
0 -I If÷
I
oll
Further, according to (4.49) we have
(XlX~ + X 2 X l ) -
(XlX] + x 3 x T ) +
(X2X% + X4X2) = 0 ,
(xIXT + X 2 X ~ ) +
(XlX~ + X 3 X l ) -
(X2X4 + X4X2) : 0 ,
(X2X ~ + x3xT) - XiX T4 - x4xT)
+ (X3X4 + X4X3) = 0 ,
(XlX~ x4x~)(x2x~+ x3x~)+ (x3x~+ x4x~, ~0 Hence, XIX T2 = _X2X 1 X3X~ = -X 4X 3 ,
T xlx~+ X3Xl ~ x2x~+ x0x~ X3X[~ XlX[ X4X~ X2X T+ 3 that is, the necessity
is proved.
It is easy to prove the sufficiency too.
(4.52)
98
4. L e t (m,m/2).
k=2 a n d H 1 , H 2 be
T h e n the r e l a t i o n s
H(n+1)
= m(xIXT
We h a v e the v a l i d i t y
(m, m/2)
4.7.
(4.42)
+ H2H I x X 2 X
n
(4.53)
Hadamard
matrices
(p, p/2)
of o r d e r
satisfy
the
(4.54)
,
(4 .55)
= 0 ,
d e f i n e d by
(4.53)
is an H a d a m a r d
(4.55)
holds
in p a r t i c u l a r ,
mat-
if
T
c)
T T HIH 2 = -H2H I
= H2H I = 0
(4.56)
xlx ~ : -x2× ~
of
(4.53)
with conditions
of H a d a m a r d
(4.42),
(4.41),
(in press)
Let
H
matrices
leads to c o n s t r u c t i o n
We w i l l give here
L E M M A 4.4.
(4.41),
(4.54)
coincide
with
S-matrices.
construction
Matevosian
(4.57) (4.58)
Xl, X 2 w i t h c o n d i t i o n s
d i n g to f o r m u l a
tions.
H(n+1)
The c o n d i t i o n
H I H ~ = H 2 H TI ,
Hence,
x x2x
mp n+1
b)
rices
T
matrix
HIH
the n o t a t i o n
in f o r m
and
HIH 2 x XIX
a)
Matrices
can be r e w r i t t e n
XI, X 2 of o r d e r
+ X2 XT = pip
4.10.
g
of o r d e r
theorem.
be i n c o m p l e t e
XIX~
T
rix of o r d e r
(4.45)
matrices
+ H I H T x XIX T2 +
of the f o l l o w i n g
Let H I , H 2
(4.41),
T h e n for e v e r y
and
+ X2X~)
a n d let s q u a r e m a t r i c e s
conditions
NOTE
(4.37)
Hadamard
= Hi(n) × X 2 + H2(n) x X 2 ,
H ( n + I ) H T(n+1)
THEOREM
incomplete
(4.42),
investigated some m o d i f i e d
be an H a d a m a r d
by r e c u r r e n t
method accor-
of HI, H 2 and XI, X 2 m a t -
(4.54)-(4.58). the case
a)
form.
It h o l d s
matrix
of o r d e r
a n d its a p p l i c a -
m
. Then there
99
exist
S-matrices
PROOF.
Let
the e l e m e n t s [I,-I]
of o r d e r
us g r o u p
of t h e s e
. Then
H
(m, m/2)
pairwise
pairs
matrix
the c o l u m n s
will
be
where der
(it is e a s y
(up to a sign)
can be r e p r e s e n t e d
H = X I ×[1,1]
to verify)
of H a d a m a r d
+ X2×
matrices
matrix;
of f o r m
then
[1,1]
,
as
[I,-I]
Xl,
X 2 will
be
S-matrices
of or-
(m,m/2). LEMMA
exist
4.5.
Let
incomplete
H
be an H a d a m a r d
matrices
H1,
matrix
of o r d e r
H 2 satisfying
m
. Then
there
the c o n d i t i o n s
T T H1H 2 = H2H 1 = 0 Proof
of the
sentation
of H
lemma
follows
immediately
from
the
following
repre-
:
HI
Jl
II
14 59~
H2
Combining
the
STATEMENT and
theorem 4.10.
lemma
4.5 a n d
the
Then
for e v e r y
n
there
lemma
matrices exists
4.5,
we o b t a i n
of o r d e r s
an H a d a m a r d
mI matrix
m1(m2/2)n
STATEMENT there
the
L e t H I a n d H 2 be H a d a m a r d
m 2 respectively.
of o r d e r
4.7,
exists
PROOF.
4.11.
Let
H 2 m be an H a d a m a r d
an H a d a m a r d
matrix
Let us n o r m a l i z e
denoting
them by H 2 × im
matrices
of o r d e r
H2m
=
the m a t r i x
of o r d e r
2m.
H2m and
divide
it into
(A0,B 0 are
2m))
H 2 x im A0 B0
Then
2m k, k = 2 , 3 , . . .
, A 0 and B 0 r e s p e c t i v e l y
((m-l),
II
of o r d e r
matrix
= Xx[1,1]
+ Y×[I,-1]
=
3parts
(-1,+1)-
100
X1
=
tf
Y
II
× [1,1]
+tl
X2
ltl
x [1,1]
,
Y2
where H 2 is an Hadamard matrix of order
2.
X
x According
YI
,x211t
,
Y =11 Y211
to lemma 4.4 matrices
X, Y
are S-matrices.
Let us introduce
the notations
Ak = Xl × Ak-1
+ Y1 x Bk_ 1
(4.60) B k = X 2 x Ak_1
+ Y2 × Bk-1
One can verify that the matrix
im
I H2mk H2mk+ I =
Ak Bk
is the Hadamard matrix of order
2m k÷l
Let us give an a l g o r i t h m of c o n s t r u c t i o n which allows to construct analogous
to Walsh [265] function
COROLLARY
NOTE 4.11. (see
systems.
Chebyshev-Hadamard
systems
There holds
4.8. Let there exists an Hadamard matrix of order
then there exists a (AT,
rix
the orthogonal
of Hadamard matrices
{2mk}~=1)
The matrix
[321]);
H2mk+ I
Hadamard
system.
is a storey-by-storey
that means that the t r a n s f o r m a t i o n
F
=
H
"
f
2mk+ I can be realized with =(NlnN)
steps
2m ,
(N=2mk+1).
Kronecker
mat-
101
THEOREM
4.8.
A[4,A,2m]
=
and an Hadamard
A[4,A,2mnk] PROOF. to p r o p e r t y ming
=
Let
Let there
exist
of t y p e
A[4,A,A,A,A,Bt,B2,B3,B4,B2m,2m] matrix
of order
2n. T h e n
there
exists
an array
of type
A[4,A,A,A,A,B~, B],B½,B~,Bk,k=2mni] H
be an Hadamard
4.2 a n d d e f i n i t i o n
S-matrices
an array
of o r d e r
Let us represent
the
matrix
4.9
of o r d e r
there
exist
2n.
Then
matrices
according
S I, S 2 for-
(2n,n) . array
A[4,A,2m]
in f o r m
AI
A[4,A,2m] =1[ A211 L e t us p e r f o r m
the p r o o f
for
B2m
= I2m
(remaining
cases
are pro-
ved by analogy).
a) A I A ~ = A 2 A ~ = i ( I m ×
4 Z B.BT i=1 I 1
T T b) A I A 2 = A 2 A I = 0 Now
let u s
introduce
the notations.
(4.61)
A 0 = S I × A I + S 2 × A 2 = A0[4,A,2mn]
It s h o w s
that
definition From find
array
definition
is the
of the a r r a y
r o w of m a t r i x
Now calculate
AoA
A0
searched
one.
Test
the
items
of
2.3.
in e v e r y
times.
the
A0
A 0 and
from construction
element
~Bi,
appears
the r e l a t i o n s
(s I x A I + s 2
A2
Is
T
T
xA I ÷ s 2
x
A T)
=
4 4 1 ) + $2S T = S I S TI x (Im × A E B i B .T 2 × (Ira× i Z B i BT) i=1 i=1
=
(4.60)
precisely
we Am
102
=
4 × AT B i B T1 } =
( S I S TI + S 2 S T ) × (I m
A n I 2 nx
Im
i=I 4
4
BiBT
= inI2mn ×
i=1
The
rows
ray of type recuired
of array
COROLLARY and Hadamard arrays
4.12.
(1971).
Let
4.13.
the m a t r i x
this process
times
A 0 ia a n a r -
we obtain
the
is p r o v e d .
e x i s t an a r r a y H [ h , k , A ] , G Z [ h , k ] , B Y [ h , k ] iI i2 ip of o r d e r nI .n 2 ....- n . Then there exist BY[h,k]
corollary
Goethals-Seidel, The
hence,
there
BX[h,k], The
. Repeating
theorem
matrices
Baumert-Hall, NOTE
The
4.9.
GZ[h,k],
NOTE
A 0 are orthogonal
A0[4,A,2mn]
array.
X B.B T i=I i i
corollary
with
4.9 g i v e s
ii
k=n I new
infinite
Wallis-Whiteman 4.9
includes
~2
n2
for
.....
classes
orthogonal ±=I
ip
np
of H a d a m a r d ,
arrays.
Plotkin
theorem
Chapter
2. C O N S T R U C T I O N
§ 5. G e n e r a l i z e d
This p a r a g r a p h general
OF G E N E R A L I Z E D
Hadamard
idea of g e n e r a l i z e d classes
existence
of g e n e r a l i z e d
number;
torable)
Hadamard
neralized
Hadamard
DEFINITION all of whose Hadamard
and
5.1.
statement
are
H*
NOTE
w i t h those
introduced
1962).
h=4t
matrix
; complex
[61]
Hadamard
H(p,p)
Note
find
in 1962. H(p,h)
is c a l l e d
matrix
h=2n;
of order h
a generalized
and H(p,p n) m a t r i c e s
Good
(1958),
that the H a d a m a r d
matrix
matrix
classic
for p=4,
of V i l e n k i n -
the a p p l i c a t i o n s
in papers
Hadamard
Rao
H(3,6)
coincides Hadamard
Kronecker
system are
respectively
(1973),
mat-
h=2t.
of g e n e r a l i z e d
(1975). also
of ge-
H.
for p=2,
Hadamard
generalized
about
matrix
of g e n e r a l i z e d
and m a t r i c e s
Trachtman
matrices.
The n o t a t i o n
matrix
(fac-
(5.1)
matrices
ces one can
is not
construc-
= hi h
Fourier
Information
about
Hadamard
by Butson
A square
p
for
of g e n e r a l i z e d
of the problem.
th roots of unity
The n o t a t i o n
of: W a l s h
rix for p=2,
provided
the t h e o r e m
generalized
is a c o n j u g a t e - t r a n s p o s e d 5.1.
H(p,h)
conditions
of
if
HH*
where
was
(Butson,
elements
matrix
of new orders;
matrices
on groups c o n s i s t i n g
of c o n s t r u c t i o n
(block-circulant)
Notations
m a t r i x construction;
some n e c e s s a r y
matrices
methods
matrices
Hadamard
matrices
matrices;
Hadamard
recurrent
tion of c i r c u l a n t 5.1.
Hadamard
of H a d a m a r d
MATRICES
matrices
survey of g e n e r a l i z e d
the known
a prime
HADAMARD
Hadamard
Berlecamp
[24]. matri(1975),
104
H(3,6)
do n o t c o i n c i d e The
with
main problem
natural
numbers
Each
p
Y
Y
X
Z
X
Z
Y
X
Y
Z
X
X
X
X
X
X
Z
X
Z
X
Y
Y
X
Z
Z
Y
X
Y
Z
Z
X
Y
Y
X
=
consist
Butson dition
p
ins o p e n matrix
In
(1964)
repeating
c a n be r e d u c e d
by elementary
H(p,h)
their multiplication
matrix
first
for
b y a fi-
row and column
problem
numbers
p
H(p,h)-matrices
problem
Hadamard
matrix
of
the n e c e s s a r y is
H(p,h)
con-
h=pt where
(to c o n s t r u c t )
t
for any p r i m e
of order
of construction
of this problem
and there relative sequences
and
Bose
h=pt
of classic
of m a x i m a l
regular
rema-
Hadamard
was
Drake (Seberry
generalized
of g e n e r a l i z e d
conducted
by Butson
(1963),
designs
(difference
sets,
length, (1969)
block-designs, relative
orthogo-
to codes
(ex-
graphs.
investigation
Later
(construction
to combinatorial
geometries)
on groups. also
to
relations)
and Goethals
strongly
1979 r e l a t i v e
called
matrix
for p r i m e
inverse
by Delsarte
codes)
matrices
The
development
matrices
Of p a r t i a l
of g e n e r a l i z e d
h=4t).
arrays);
tended
matrix
of n o r m a l i z e d
like well known
Shrikhande
5.1.
commutation,
that
a generalized
Further Hadamard
Hadamard
proved
number.
(p=2,
of u n i t y ,
.
to a n o r m a l i z e d
for e x i s t e n c e
number
h
3th r o o t s
I.
(1962)
is a n a t u r a l
linear
of
from note
is c o n s t r u c t i o n
(row a n d c o l u m n
x e d r o o t o f unity) which
matrices
and
generalized
operations
were
X
/~ , Z = - yi - i --~ /5 are X = I, Y = - ~1 + i --~
where
nal
Z
of
h-geometries
introduced 1979,
Hadamard
(generalization
the n o t a t i o n
Street
matrices
1979)
of H a d a m a r d
these
on groups.
matrices
105
DEFINITION a square of
5.2.
matrix
finite
GH(r,G)
group
G
-Ik}
{hi,kh j G same
with
number
Denote ty and by Drake
(Drake, =
1979).
(hij)
of o r d e r
I ~ k ~ r
every
b y Kp the yp
multiplicative
the original
(1979)
noted
that
the
Kp,
are prime
numbers.
5.1.
of g e n e r a l i z e d damard
Kq)
If
consists
l e t us
of all elements
group
notation
is n o t matrices
GH(r,Kq)
a prime H(q,r)
on group
show that
=[hi, j ] , t h e n
i#j the every
G
is
of e l e m e n t s sequence
element
of g r o u p
p
th r o o t s
of u n i -
of generalized matrix
number,
Hadamard
GH(r,Kp)
then
mat-
on group
the n o t a t i o n
strictly
that
of H a -
Kq.
from definition
from Kq
of a l l
includes
if H = G H ( r , K q )
, then
5.2 a n d
H=H(q,r) . Let H=GH(r,
from the'fact
that
the
sum
is 0, we h a v e
r Z h . . h -1, k=1 l,n 3,~ Hence,
consisting
if for
that of Hadamard
q
Hadamard
matrices
First
on g r o u p
root of Kp
with
STATEMENT
time
2
matrix
times.
coincides
p
of o r d e r
p(IGl = p),
rix H(p,r) if
An Hadamard
= c
q-1 1 z yq = 0 i=0
H=H(q,r).
L e t us n o w g i v e not a GH(r,Kq)
h(6,10)=
an e x a m p l e
of n o r m a l i z e d
matrix
H(q,r)
which
is
matrix. i
I
I
I
I
I
I
I
I
X4
XI
X5
X3
XI
X3
X3
X5
XI
X1
X2
X3
X5
X5
X1
X3
X5
X3
X5
X3
X2
X1
x5
X3
x5
X3
X1
X3
X5
X1
X4
xI
x1
X5
X3
X3
I
1
X3
X3
X3
x3
X3
1
1
XI
XI
X5
X3
X4
x3
I
X2
X4
XI
X5
x3
X5
x2
X4
X3
X2
I
X5
X3
X5
XI
X2
I
x2
X3
X4
(5.2)
106
It is c l e a r plete
that
generalization
Hadamard
matrices
this class
(Drake,
- commutation of GH-matrices
- if
the G
since
matrices of c l a s s
they
are
not com-
of g e n e r a l i z e d
do n o t c o i n c i d e
with
of g e n e r a l i z e d
Hadamard
matrices
1979): of r o w s
or c o l u m n s
property
in f i r s t
and multiplication
by a fixed
of generalized group,
element
Hadamard
then one
row and columns
from center matrix
can get
of w h i c h
the e l e m e n t s of group
GH on g r o u p
Hk matrices
G;
on
is a u n i t e l e m e n t
G
of
G.
truction
propose ons with
after
matrices
new methods orthogonal
Let us now give matrices
H(p,h)
on groups;
tables the
constructed
p
(Butson,
1962);
H(2,h)-matrix - H ( p , p k) a non-negative
with
- H(p,p2((p-1)t+1)) rameter
o f an o r t h o g o n a i
- Symmetric is a n a t u r a l - circulant
is a n y n a t u r a l
(Butson,
array
H(2k,4k)
core
the c o n s -
on g r o u p s
D.Jungnickel and
their
and
(1979) relati-
configurations.
of generalized
matrices
GH(r,G)
Hadamard on groups:
natural
num-
1962); nulaber,
where
p
h
is the o r d e r
is a p r i m e ,
k
of
is
1963);
where
and regular number
classes
(Butson,
the c i r c u l a n t
integer
(1979),
m ~ k, k is a n a r b i t r a r y
integer
, where
matrices
combinatorial
Hadamard
p,
introduced
of G H - m a t r i c e s
and other
, for p r i m e
(1979)
Hadamard
J.Street
of c o n s t r u c t i o n
is n o n - n e g a t i v e
- H(2p,h)
Y.Seberry
of g e n e r a l i z e d
and generalized
- H ( p , 2 m . p k) m
D.A.Drake,
of n e w c l a s s e s
of w e i g h t e d
m
s
some properties
row or column
Immediately
ber,
GH(r,G)
of g e n e r a l i z a t i o n
for e v e r y
is a n a b e l i a n
each element group
by Drake
for G = K s.
on groups
retain
of c l a s s
H(s,r)
L e t us n o w n o t e
G
introduced
p
is a p r i m e
OS(s,t)
H ( p , p 2m)
(Delsarte, , where
(Shrikhande, , where
Goethals, k
number,
p
t
is t h e
pa-
1964);
is a p r i m e
number,
1969);
is a n y n a t u r a l
number
(see t h e o -
107
rem
5.6) ; H ( 2 p , m k n)
-
the o r d e r
, where
of a h y p e r f r a m e
H(2p,21n),
-
m
where
(see t h e o r e m n
5.5) .
(Belov
is an a b e l i a n
- G H ( p i+j
Cpi)
for
- GH(p2(p2-1),Cpr) berry,
of H ( p , m ) - m a t r i x ,
group
all
of g e n e r a l i z e d
of o r d e r
i > I, j > 0
, where
k
is
5.2);
is the o r d e r
(see t h e o r e m Cpi
is the o r d e r
p
i
, p
(Drake,
p2 and
Yang matrices
is a p r i m e
number).
1979);
pr-1
are p r i m e
powers
(Se-
1979) ;
- G H ( 2 p ~, G F ( p ~ ) * ) , - if p
r-1
= q
s
G F ( p ~)
for
some
- is G a l o i s
prime
number
field
(Jungnickel,
q, t h e n
there
exist
1979); (Street,
1979) ; a) G H ( p r k + t ( p r - 1 ) b)
, Cpi)
for e v e r y
GH(2mpk+t+ri(pr-1)J,cp)
I< i < r
for e v e r y
, I _< j _< k
0 <m
, t > 0.
k>_ I, i > j > I,
t>0. C)
G H ( 2 m p ek,
L e t us matrices
first
Cp ~ ) for e v e r y
formulate
introduced
in the m a i n
I. If H 1 = H ( P l , h ) the P3
P2 o r i g i n a l = L'C'M-(PI'P2) 2 ' If H =
matrix,
some
0 <m<
properties by B u t s o n
is a g e n e r a l i z e d
root
of unity,
(L.C.M.
(h i ,j)i,j h
then
is the
= I
k, k >
I.
of g e n e r a l i z e d (1962) ;
Hadamard H 2 = YP2HI
least
Hadamard
matrix
a n d y P2
= H(P3,h),
common
multiple).
is an g e n e r a l i z e d
normalized
is
where
Hadamard
then
h Zh. j:l l'J
=
h lh~ j:1 l'J
= 0,
i=2,3,...,h
(5.3)
h Z h. i=I l,j
=
h Z h c . = 0, i=I 1'3
i=2,3,...,h
(5.4)
108
that
is the
sum of e l e m e n t s
row and first column
of a n y r o w a n d any c o l u m n e x c l u d i n g
is zero.
3. If H I = H ( P l , h I) a n d H 2 = H(P2,h2) matrices,
(where P3 = L ' C ' M ' ( P I ' P 2 ) ' with parameters
P2 and
L e t us d e n o t e by
F
tive
r o o t of unity;
p-th
is t a k e n m o d u l o
h3=h1"h2
~ is a n a t u r a l
p-th roots
group relative
subgroup
g r o u p order,
•
subgroups
(r)
According ments
denote
by
yp
q
n-th primi-
we w i l l m e a n t h a t the p o w e r
of u n i t y
S
in f i e l d of c o m p l e x n u m b e r s
to m u l t i p l i c a t i o n
GI,G2,...G
a n d are g e n e r a t e d
by
_I
(non-trivial) is a l s o c y c l i c
subgroups and
f r o m q -th, q 2 - t h , . . . , respectively
of g r o u p
of g r o u p
its o r d e r
di-
q~-1-th
roots
F:
2Uj. }q2-I = e x p ( i ----~ q j=0 , r = I , 2 , . . . , ~ - I .
G
let us c o n s t r u c t
ajacent
classes
F:
ymk q~
group
t h e n we have
to s u b g r o u p s
=
number;
where
2Us }q~-1 e x p ( i - - ~ } s=0
in p r o p e r
of a c y c l i c
= f jqe-r
of g r o u p s
x(k) mk
matrix
p - t h r o o t of unity.
We w i l l be i n t e r e s t e d
G r =%qj
Hadamard
p.
It is k n o w n t h a t
every primitive
(5.5)
is a g e n e r a l i z e d
in w r i t i n g
s F = {fs = Y q a =
f o r m the c y c l i c
,
a g r o u p of q e t h r o o t s of unity,
number,
of u n i t y
Hadamard
h3 .
is a p r i m e
vides
are g e n e r a l i z e d
then
H = H I × H 2 = H ( P 3 , h 3)
F. Since
first
= ' Gk
ink=l,2,..., q e - k - I ,
{f jq ~-k
+m k
= ¥ 3.q ~-k q
k = I , 2 , . .., ~-I
+ mk
} qk-1 j =0
of e l e -
109
Constructed wing
X (k)
sets
, mk=1,2,...,qa-k-1
, satisfy
follo-
the
conditions : a)
k-1 =q U -I X(1) r =0 mk+rq~-k
X (k)
where
b)
k=2,3,...,e-1,
the
mk=0,1,...,qe-k-1
s u m of all e l e m e n t s
from each
-I
that
q-1 Z fjq~-I = O. 9 =0 +n
H=H(q~,h)
be a n o r m a l i z e d
~-I q
, and
is
zero,
set
X~ k)
X(1! n
= Gk
n=0,1,2,...,
It is true THEOREM matrix, rows
q
5.1.
Let
be a p r i m e
(columns)
number,
of m a t r i x
mk=l,2,...,qa-k-1
H
~ be a n a t u r a l include
and G1,G2,...,G
Then
h=qt
where
t
5.2.
An a l g o r i t h m
only
generalized
number;
elements
of
Hadamard
suppose set
that
X (k) mk
the
r
_1
is a n a t u r a l
number.
of c o n s t r u c t i o n
of g e n e r a l i z e d
Hadamard
matrices
Let us c o n s i d e r mard
matrices
an a l g o r i t h m
of n e w o r d e r
of c o n s t r u c t i o n
which
allows
of g e n e r a l i z e d
fo f a c t o r i z e
obtained
Hada-
matri-
ces. DEFINITION
5.3.
Square
of 0 and
p-th
roots
ting
matrices of u n i t y
X and
and
Y
of o r d e r
satisfying
the
k consis-
following
con-
ditions: X + Y
is the m a t r i x
consisting
of p - t h
roots
of u n i t y .
X * Y = 0 XY*
= YX*
XX*
+ YY*
is c a l l e d
(5.6) = kI k
a generalized
Let us d e n o t e
2-elemental
it by G(p,k)
hyperframe
= {X(p,k),
of o r d e r
Y(p,k) }.The
k.
notation
of ge-
110
neralized mental
2-elemental
hyperframe
STATEMENT H(p,2m)
5.2.
THEOREM
If there e x i s t
suppose
a n d G(p,h)
= {X,Y}
be a
t h a t the f o l l o w i n g
and
n
H' n
H*
O0
=
generalized
be a
matrices
2-elemental
+
generalized
generalized
condition
H'H*
=
O0
determined
Hn
Hadamard
hyper-
respectively.
H 0 = H(Pl,m)
H
Then H
for p=2 w i t h that of 2 - e l e -
generalized
then there exist
5.2. L e t
r i x a n d G(P2,k)
coincides
[4 ].
a n d H(p,h)
f r a m e s G(2p,2m)
hyperframe
I
+
mat-
hyperframe;
holds:
0
(5.7)
by r e c u r r e n t
X x Hn_
2-elemental
Hadamard
relations
Y x H'
n-1
(5.8) H n' = X × H'n_1 - Y x Hn-1
are: I) g e n e r a l i z e d if G . C . M ( P l , 2 )
Hadamard
matrices
= I and G.C.M(P2,2)
H ( 2 p , m k n) w h e r e p = L . C . M . ( p l , p 2)
= I (G.C.M.
is the g r e a t e s t
common
measure). 2) g e n e r a l i z e d
Hadamard
matrices
H ( p , m k n) w h e r e
p= L.C.M. (pl,P2) . NOTE
5.2.
lar by the
H ~
o
=
(Im/2 × II0 -I
al,
H °' can be r e p r e s e n t e d
I
II )H o
, where
0
in p a r t i c u -
ez " P2
H
= H(Pl,m)
is a m a t r i x .
o
5.3. Let p l , P 2 , . . . , p I
an g e n e r a l i z e d
al H(2Pl
where
5.2 m a t r i x
formula
STATEMENT exists
In t h e o r e m
Hadamard
Then there
matrix
~ BI ..... pl I, mk I
~z,...,~i,61,
be p r i m e n u m b e r s .
62,...,
B2 k2
Bl-1 ...." kl_ I
BI_ I are n a t i r a l
numbers.
111
THEOREM represented
5.3.
Defined
as p r o d u c t
in t h e o r e m
of w e a k l y
Hn = M I
M2
M n + I = Ikn
5.2 m a t r i x
filled
of
form
m a t r i c e s [194]
Pmkn_ 1 = Imkn_i/2
..." M n + I ,
x H°
× II 0
,
(5.9)
111
-I L e t us n o w the classic
introduce
Yang
DEFINITION order
n
5.4.
of
AA*
will
be c a l l e d Note
that
coincides
generalized for
with
p=2
that
search
A =
From
(5.10)
, 1 = 1 , 2 ..... n.
of Y a n g m a t r i c e s
generalized
square
p-th
+ BB*
roots
= 2nI
matrices
A(p,n)
a n d B(p,n)
satisfying
of
the c o n d i t i o n
(5•10)
n
Yang matrices. of g e n e r a l i z e d
of Y a n g m a t r i c e s construction
the m a t r i c e s
n-1 Z a . U 3-I 3=0 3
and extend
case.
of u n i t y
the n o t a t i o n
L e t us n o w c o n s i d e r We w i l l
to the
Circulant
consisting
,
0
the n o t a t i o n
theorem
c a n be
:
M 1 = Iml_1 x (X x I m k n _ 1 + y x P m k n _ l )
where
(5.8)
A
and
B =
t
Yang matrices
[324]. of g e n e r a l i z e d B
Yang matrices.
in f o r m
n-1 Z 1 Uj j=0 3
we g e t
n-1
5j=0
--
-
(a3a (3-t) m o d n
It is e a s y condition
5.4.
generalized
H(2p,2i+In)
to see
3
5
(j-t) m o d n
that
for e x i s t e n c e
THEOREM are
+b
(5.11)
) = 0,
t=1,2,
.... [ n / 2 ]
is the n e c e s s a r y
of g e n e r a l i z e d
Hadamard
(Generalized
Yang
theorem).
Yang matrices
then
there
If
exist
and
(5.11)
sufficient
matrices• Ao{P,n)
and Bo(P,n)
generalized
matrices
112
DEFINITION circulant
5.5.
The
generalized
Hadamard
matrix
H=H(p,h)
is
if
H = aoI
+ aiU
where
aj,
same)
if the r e l a t i o n
+ a2 U2
+...+
, are
p-th
j=0,1,...,n-1
a n_1Un-1
roots
of u n i t y
or
(what
is the
n-1 X aka (k-t) m o d k=0
= 0,
n
t=1,2 ..... [n/2]
(5.121
is true. DEFINITION
5.6.
the n o r m a l i z e d and a column Let
generalized (all
us give
a)
b)
there
STATEMENT
of g e n e r a l i z e d
1
I
I
I
I
XI
X2
X2
XI
I
XI
I
XI
X2
X2
I
X2
XI
I
XI
X2
1
X2
X2
X1
I
X1
I
XI
X2
X2
XI
I
obtained
by d e l e t i n g
of H(p,h)
matrix
I
roots
with
from
a row
matrix. circulant
core:
exists
by B u t s o n
(1963).
whose
to the e x i s t e n c e
rows
form group
of an
and
m-sequence
on
pU-1. an
m-sequence
an o r t h o g o n a l
n=p-1, 5.4.
of unity.
obtained
is e q u i v a l e n t
exists
m=(pU-1)/(p-1),
the core
of a H ( p , p U ) - m a t r i x
the p e r i o d
If t h e r e
be c a l l e d
h-1
H(p,h)
I
3-th
The e x i s t e n c e
with
matrix
I
two r e s u l t s
(circulant)
GF(p)
then
also
H o of o r d e r
=
I, X I, X 2 are
Give
core
I) w i l l
matrix
Hadamard
an e x a m p l e
H(3,6)
where
A square
k=p u-l,
There
exist
of GF(p)
difference
with
the p e r i o d
set w i t h
parameters
pU-1
6=pU-2 block-circulant
generalized
Hadamard
113
m a t r i c e s of following parameters.
H(P3,4n), where
m
H (P2,m)
H(P4,6n),
H(P5,8 n)
is the order of the c i r c u l a n t g e n e r a l i z e d H a d a m a r d m a t r i x and
P3 = L.C.M(2Pl,P2), Dubenko,
P 4 = L . C . M ( 3 P l , P 2 ), P5 = L . C . M ( 4 P I , P 2)
Zaitzev had p r o p o s e d the c i r c u ! a n t g e n e r a l i z e d H a d a m a r d
m a t r i c e s H(2k,4k)
for any natural number
k
.
Let us give another proof of this theorem. T H E O R E M 5.5. There exists a c i r c u l a n t g e n e r a l i z e d H a d a m a r d matrix H(2k,4k)
for any n a t u r a l number
k.
PROOF. We will c o n s t r u c t the matrix H=H(2k,4k)
as follows.
First
row of the m a t r i x c o n t a i n s in even places the e l e m e n t s of main diagonal of side V a n d e r m o n d e m a t r i x V = H(2k,2k), in odd places rix
V=II Vi,jl I ={yk i'j} and
it c o n t a i n s the e l e m e n t s of side from diagonal of mat-
V. R e m a i n i n g rows of
H
are o b t a i n e d first row by c i r c u l a n t
shifts. Let us prove that the c i r c u l a n t matrix T r a n s f o r m the matrix
H
to the m a t r i x
H
is a H(2k,4k)
matrix.
H' by c o m m u t a t i o n s of rows
and columns a r r a n g i n g them in the f o l l o w i n g order. E q u i v a l e n t to
H - m a t r i x o b t a i n e d is of b l o c k - c i r c u l a n t
II
H' =
A
AI
o A~
Ao
V
rows. H'H'*
first rows are the ele-
side A~ d i a g o n a l s respectively.
the c i r c u l a n t m a t r i x o b t a i n e d from
Prove that
(5.13)
II
where A ° and A I are c i r c u l a n t m a t r i c e s whose ments of main and
= 4kI4k
,
form:
AI
A~-matrix
is
by the c i r c u l a n t shift of
114
AoAI* ÷ AIAo* = 0 , AoAo* + AIAI* = 4kI2k , (5.14)
A~A~* + AoAo* = 4kI2k
The first equation of system
(5.14) follows from AoA{* = AIAo*,
sin-
ce
2k-lj =0Y2kZ (J-tl)2 + (J-t2)2+j-t2
t12+t~-t 2 =
Y2k
k-1 (
= Y2kt12+t2-t22
2j2-j(1-2t1-2t2 )
Z
k-1 -
j=0 Y2k
2k-lj:0Y2kZ 2j2-j(1-2t1-2t2 )
2j2-j(1-2tl-2t2)
~
)
j=0 Y2k
Two remaining equations of system
(5.14) are also true since
2k-I -(J-tl)2-j+t1+(J-t2)2+j-t2 Z T2k j=0
=
x
J-t2-1 y2 k
=
0
2k-1 (J-t1-1) 2-j+t1+1+ (J-t2-1) Z T2k j=0
2k, t1=t 2 ={ 0 , t1~t 2
Hence, the circulant matrix
H
is a H(2k,4k)
§ 6. Construction of Hi~h-dimensional
matrix.
Hadamard
matr ice s
In this paragraph we will give a review and two approaches of construction of high-dimensional
(proper and improper)
Hadamard mat-
rices. First of these approaches is new one and the second extents the known "flat" methods to the high-dimensional
case and gives so-
lution of Shlochta problem. Later we will introduce the notations of density, weight and surplus for (flat and spatial) Hadamard matri-
115
ces.
Upper
and
lower
tion
of h i g h - d i m e n s i o n a l
of c o n s t r u c t i o n generalized
estimates
H ighTdimensional
From
1970
coding
theory
and
in i m a g e
in h i g h - d i m e n s i o n a l that
so the
typical
model
three-dimansional
The b l o c k sent
atoms Before
give
matrices
proper
Hadamard
found.
Nota-
and a l g o r i t h m s
high-dimensional
matrices.
application ~,
shown.
can
crystal
I.
atoms
of
be
(see
of o r d e r
Fig.
of H a d a m a r d
matrices
225 ] a d e f i n i t e
was
matrices
matrix
are
introduced.
matrices
of r o c k - s a l t
represent
Hadamard
processing
Hadamard
Hadamard
circles
with
Hadamard
high-dimensional
are
classic
in c o n n e c t i o n
introduced
and p a r t i a t l y
matrices
6.1.
values
generalized
of c o m p l e t e l y
Hadamard
for
in
interest
It is n o t e w o r t h y
found fig.l)
in n a t u r e
too;
is a p r o p e r
4.
sodium
and w h i t e
circles
repre-
of c h l o r i n e . definition
of h i g h - d i m e n s i o n a l
Hadamard
matrices
we w i l l
some e x p l a n a t i o n s . Let
us h a v e
from n p elements which
are
dinates
some n u m e r i c a l ail
arranged
field
It is k n o w n
that
. ..i ' i 1 ' i 2 ' ' ' ' ' i p = 1 ' 2 ' ' ' ' ' n ,12," p,
in the p o i n t s
il,i2,...,i p
f.
, form a
of p - s p a c e
p-matrix
, of
determinated
of o r d e r
n
any
by
system
field the
on f i e l d
p,
coorf.
116
A =II A i l , i 2 , . . . , i p II The t o t a l i t y is any n u m b e r which
is the
Using
from
(p-1)-matrix
(for the
depending
For e x a m p l e , entations
11
with
1,2,...,p,
2-dimensional
onal matrix table
of e l e m e n t s
,
(il,i2,...,ip=l,2,...,n)
fixed value
f o r m the s e c t i o n
of o r d e r
sections
n
one c a n w r i t e
the c u b i c m a t r i x of o r d e r
AI 11
A112
A121
A122
1
where
of o r i e n t a t i o n
(i s)
d o w n the h i g h - d i m e n s i -
b y a s q u a r e or r e c t a n g u l a r
the d i m e n s i o n
(i) can be r e w r i t t e n
is
.
sake of o b v i o u n e s s )
on w h e t h e r
of i n d e x
(6.1)
of m a t r i x
2(p=3,n=2)
is e v e n or odd. by s e c t i o n s
of o r i -
in f o r m of r e c t a n g l e
A211
A21211
~
(i)
A221
A222
~
(k)
(6.2)
+ (j)
(arrows
indicate
the d i r e c t i o n s
in w h i c h c o r r e s p o n d i n g
indices
are
increased). DEFINITION der
m
(j),
two-dimensional
(k) s a t i s f y
layers
=
m
Hadamard
matrix
in all axis n o r m a l
of or-
if all
orientations
z ha,j,khl,j, k = j,k
2 = l,K Z h i , a , k h i ,b,k = m 6 a , b
DEFINITION
, hi,j,k=~1
the c o n d i t i o n s
h. h. i,j l , j , a 1,j,b
der
H =II hi,j,kll
w i l l be c a l l e d a t h r e e - d i m e n s i o n a l
its p a r a l l e l (i),
6.1. A c u b i c m a t r i x
'
6.2. A c u b i c m a t r i x
w i l l be c a l l e d a r e g u l a r
two-dimensional
layers
are t h e m s e l v e s
Hadamard
0, a~b 6a,b = I, a = b
H =If h i , j , k l l ,
Hadamard
in all axis n o r m a l matrices.
matrix
(6.3)
h i , j , k = ~ I of or-
if all its p a r a l l e l
orientations
(i),
(j),
(k)
117
That
is:
Zh. • l,a,rhi,b,r l
=
~ha, j , r h b , j , r 3
= m6 a,b
(6.4)
Zh i i 'q'ahi'q'b
=
~ha,q,khb,q,k
= m6 a,b
(6.5)
Ih . i P'3'ahp'j'b
=
Zh k P'a'khp'b'k
= m6
(6 6) °
for all
r-values
Conditions hold.
Reverso
of p a r a m e t e r s
(6.3)
k,q,p
are a l w a y s
statement
a,b
.
satisfied
is not a l w a y s
if
(6.4),
true w h i c h
(6.5),
(6.6)
are
shows the f o l l o w i n g
example:
-I
I
II 1
Fig.
2 shows cubic
s e c o n d one
I
-1
1
1
Hadamard
II
matrices.
First
one
is r e g u l a r
the n o t a t i o n
of o r t h o g o n a l i t y
I
bI
ai=
i i ,a~) (al,a2,...
Ia 2
b2
bi=
i i i ( b ] , b 2 , . . . , b n)
and
= ,
of two m a t r i c e s
B
(6.8)
=
i=1,2,...,n i " i
A'B= an
matrix,
B
i a A
(6.7)
is not.
we i n t r o d u c e
If
A
I
1
bn
n aibi i=I
=
0
118
where
aib i
is a s c a l a r
can be r e f o r m u l a t e A cubic will
nal
matrix
be c a l l e d
layers,
as
product
of two v e c t o r s ,
then
definition
6.1
follows.
H =II hi,j,kll
the H a d a m a r d
in all a x i s - n o r m a l
, hi,j,k=t I
matrix
if all
orientations
of o r d e r
parallel
(i),
(j),
m
on
two-dimensional (k)
are o r t h o g o -
in p a i r s . Note
in all
second
definition
demands
the o r t h o g o n a l i t y
of r o w s
sections.
In onal
that
1971Shlochta
Hadamard
matrices.
(cubic)
Hadamard
duct
3
of
for the In
matrices
in d i f f e r e n t
first
[225] based
time
constructed
Andrews on the
directions
3- and
constructed
construction
(i),
(j),
4-dimensi-
inproper of d i r e c t
(k) o r i e n t e d
statial pro-
Hadamard
matrices. Now matrix
let us r e p r e s e n t of o r d e r
orthogonal
construction
4, g e n e r a t i o n
Hadamard
matrices
by d i r e c t of o r d e r
Fig.
In
1979
Shlichta
high-dimensional found
class
introduced
Hadamard
of p r o p e r
of
E q
I r
Hadamard
of t h r e e
mutually
3.
of o r d e r
matrices)
. .. T h p , q , r , . . . , y , a y
cubic
2.
II hi,j, k ..... m II ' h i , j , k ..... m = -+I w i t h
E p
product
an a l g o r i t h m
matrices
Hadamard
irregular
that
for c o n s t r u c t i o n k=2 n is
of
(till n o w the o n l y m-matrices
the c o n d i t i o n
" hp,q,r,...,y,b
= Km-1~
a,b
(6.9)
119
where
the s u m m a t i o n
dinates
covers
all c o m m u t a t i o n s
(p,q,r,...,y,z)
of coor-
(i,j,k,...,m).
Fig.
4 shows
nal H a d a m a r d
the e x a m p l e s
of proper
and
improper
four-dimensio-
matrices
~v Fig.
The author Williamson existence
of
matrix
(1,1)
matrices.
3
orthogonal
, (1,1,1)
the t h e o r e m
for
investigation
on,
1979); b) c o n s t r u c t
nal H a d a m a r d
the n o t a t i o n orthogonal
high-dimensional
of h i g h - d i m e n s i o n a l
some
Hadamard
analogous
of
designs
weighted
matrices
we-
matrix,
matrices
is rather
questions:
case
an a l g e b r a i c a l
to 2 - d i m e n s i o n a l
high-dimensional
of order
Hadamard
unsettled
for h i g h - d i m e n s i o n a l
question,
mard m a t r i x
introduced
investigated
of
matrices.
and c o n s t r u c t e d
of h i g h - d i m e n s i o n a l
analogoues
case
apparatus
(Shlichta
of W i l l i a m s o n
questi-
matrices
1973);
for any n a t u r a l
6.1)
and
Let us p r e s e n t
a) c o n s t r u c t
(Shlichta
3
about e q u i n a l e n c e
Williamson
in 1979
designs
of h i g h - d i m e n s i o n a l
there.
investigation
incomplete.
tion
the n o t a t i o n
and J . S . W a l l i s
Four a p p l i c a t i o n s
re also g i v e n
c)
and f o r m u l a t e d
Hammer
high-dimensional
The
introduced
of flat and h i g h - d i m e n s i o n a l
Finally
of type
[8]
4.
number m
. Note
then
the n e c e s s i t y
m
construct
that
m~0(mod
if
m
4) but
of the c o n d i t i o n
a high-dimensional is a r e g u l a r
in general m~0(mod
Hada-
high-dimensio-
case
(see defini-
4) remains
open.
120
Let
us
matrices.
now
consider
the
Introduce -c
it-d-c
-cd
ba -ba
b -a
d -c
d
d
PB (a,b, c,d)
construction
of h i g h - d i m e n s i o n a l
Hadamard
badilbcd i
a -b
-a -b
c -b -a
-a -b -c -d
-d
-c
-c
-d
b
-b
a -c
a -d d
c
a -b
~(k)
li
li a
c
-b
a -d
c
c -d -a
-c
b
a +(j)
the
three-dimensional
PY(a,b)
analogue
=
V
0
=
us
introduce
the
Williamson
-b
a
a
b
-a
-b
-b
a
the
Yang
analogue the
of
following
array
(6.11)
array
[304].
notations
1 0 0 . . . 0 0
010
...
0 0
0 0 0 . . . 0 0 1
0 1 0 . . . 0 0
001
...
0 0
1 0 0 . . . 0 0 0
0 0 1 . . . 0 0
[304].
I/
II
is a t h r e e - d i m e n s i o n a l Let
of
•
•
.
~ . o
.
•
•
•
•
• o .
.
.
0 t 0 . . . 0 0 0
000...I0
000
...
01
000...I00
0 0 0 . . . 0 1
100
...
00
000...000
= II II u I 21... l n-111 , un-1 Vl
=
i U
I I "'"
Iun-2
1 .....
Po
= PB(-I,-I,-I,-I),
PI
= PB(-I,-I,-1,1)
' P9
= PB(1,-I,-1,1)
,
P2
= PB(-I,-1,1,-I)
, PI0 = P B ( I , - 1 , 1 , - I )
,
P3
= PB(-I,-1,1,1}
P4
= PB(-1,1,-I,-I)
P5
= PB(-1,1,-1,1)
,
P13 = P B ( 1 , 1 , - 1 , 1 )
,
P6
= PB(-1,1,1,-I)
,
P14 = P B ( 1 , 1 , 1 , - I )
,
P8
,
Vn-1
=
= PB(I,-I,-I,-I~
P11 = P B ( I , - 1 , 1 , 1 )
un-2 I
i''"
II I
,
,
, P12 = P B ( 1 , 1 , - I , - I )
(6.12)
,
121
P7 = P B ( - 1 , 1 , 1 , 1 )
Investigate
,
n o w by a n a l o g y
P15
with
= PB(1,1,1,1)
plane
case
the
following
question.
Let
HHmn
What
conditions
der
n
to
HH
= V ° x H ° + V I x H I + ...
one
must
lay u p o n
the m a t r i c e s
be a t h r e e - d i m e n s i o n a l
mn
(6.13)
+ V m _ 1 x Um-1
H o , U 1 , . . 0 , H m _ 1 of or-
Hadamard
matrix.
It is true STATEMENT and
6.1.
In r e p r e s e n t a t i o n
H i = Hm_ i, H i 6 { P o , P I , . . . , P 1 5 } Then
(6.13)
let
m
be an o d d n u m b e r
, i=0,I,2,...,m-I.
if
H 0 6 {Po,P3,P5,P6,P9,P10,P12,P15}
= p1
H i 6 {P1,P2,P4,P7,Ps,P11,PI3,P14}
= p2,
so
and
vice
versa
if
It is a l s o THEOREM exists there tion
a proper
(6.5),
If t h e r e spatial
a regular then
there
Let us n o w c o n s i d e r DRFINITION
Aft
= {apq}
be c a l l e d
6.3.
a Williamson layers
2-dimensional NOTE
6.1.
exists
Hadamard Hadamard exists the
Cubic
, apq=~1,,
two-dimensional are
H i 6 p1 , i : I , 2 , . . . , m - I .
true
6.1.
exists
H ° 6 p2 ,
i=1,2,...,m-I
a BCBSH matrix matrix
a BCBSH
solution
matrices
f6{it,Jt,kt}
of o r d e r
4m
of o r d e r
4m.
of o r d e r of o r d e r
4m
, n=1,2,...,I
{AI,A2,...,AI,Im,m}
in all
axis-normal
with
versa
if
representa-
problem.
II Ai,j,kll
family
Vice
there
4m.
of S h l i c h t a An =
then
= {Ait,A~ ,A. } m 3t K t t=l of o r d e r if all
orientations
families
of W i l l i a m s o n
matrices.
The n o t a t i o n
of W i l l i a m s o n
families
m
will
its p a r a l l e l
(i) ,(j), (k)
coincide
(as in 2-
122
dimensional for 1=2
case)
that of spatial
it is a "spatial"
"spatial"
analogue
STATEMENT rices
with
of 8 W i l l i a m s o n
6.2.
Foe e x i s t e n c e
{AI,A2,...,AI,Im,m}
of 2 - d i m e n s i o n a l From the
family
statement
three-dimensional THEOREM 1=2,4,8,
6.2.
analogue
of a n a l o g o u s
Williamson
matrices
there
exists
a cubic
1=8
it is a
type W i l l i a m s o n
of order
a cubic
mat-
the e x i s t e n c e
matrices.
the e x i s t e n c e
of
m, m6L I.
family
and spatial
regular
of W i l l i a m s o n
and sufficient
P B ( X l , X 2 , . . . , X k) k=4,8, Then
family
in p a r t i c u l a r
matrices
Let there exist
for
for 1=4,
matrices.
of special
follows
matrices
of Yang matrices,
it is n e c e s s a r y
6.2
of W i l l i a m s o n
Williamson
{AI,A2,...,AI,Im,m},
arrays
PY(YI,Y2)
Hadamard
matrix
of order
l'm,
1=2,4,8. NOTE
6.2.
The
theorem
(1979),
Williamson
spatial
case.
Note
struct
the
to e x t e n d
dimensional Note
Baumert-Hall
case
6.2.
will
and
one
hyperframe
to the
the n o t a t i o n
in turn,
of Baumert-Hall,
allows
theorems
of
to con-
Goethals-Sei-
Goethals-Seidel
and G o e t h a l s - S e i d e l
of Yang
respectively
introduce
which
Baumert-Hall,
case with certain
of H a d a m a r d
method
arrays
hen-
to the high-
density
here
of H a d a m a r d
(classical
and lower b o u n d
density
and
matrices
the n o t a t i o n s
matrices
can also be e x t e n d e d
restrictions.
lower bounds . for w e i g h t
introduce
find upper
theorems
Paley-Wallis-Whiteman
excess
We will
6.2
the e x t e n s i o n
too.
also that
upper
(1976)
analogues
to e x t e n d
to h i g h - d i m e n s i o n a l
density
statement
a 2-elemental
hence,
is for 1=2,4,8
Wallis
the h i g h - d i m e n s i o n a l
del arrays ce,
(1944),
that using
(and construct)
6.2
of weight
density
and e x c e s s
and t h r e e - d i m e n s i o n a l )
of these
values.
and
123
DEFINITION of o r d e r
n
6.4.
n
6.5.
plane
the
, i=2,2
1's
W ( H n)
in H
A excess
n
following
of the
Hadamard
matrix
Hn
.
0(H n)
sum of all e n t r i e s
is the
(classical)
of
[217].
is the
Introduce H (i) n
A weight
is the n u m b e r
DEFINITION of o r d e r
[217].
of the
of m a t r i x
Hadamard
matrix
Hn
H n.
notations:
set of all H a d a m a r d
matrices
for
i=2
matrices
of o r d e r
and of t h r e e - d i m e n s i o n a l
n
ones
(of for
i=3) ; [H~ i) ] , i=2,3 order
n;
rices
n
n;
matrices
of o r d e r
be c a l l e d
that
[37~217]
= W ( H n) bound W(n) In
1977
n
a maximal
mat-
set of all
Hadamard
matrices
be c a l l e d
a weight
of Ha-
.
excess
of H a d a m a r d
matrix
of o r d e r
n
.
= °(i) ([HBn(i) ]) = m a x { ° ( H n ) ;Hn6[ HB n(i) ]}
Schmidt
had raised
for a g i v e n for W(n)
n
a question
. The p r o b l e m
is c o n s i d e r a b l y
is d e f i n e d Schmidt
will
= m a x { o ( H n ) ; H n C [ H ( i ) ]} n
o(i) ([HBn])
upper
Hadamard
of
= m a x { W (i) ( H n ) ; H n 6 [ H B n (i) ]}
0(i) ([Hn])
W(n)
is the
type
matrices
(i) ] } = m a x { W (i) (Hn) ;Hn6[H n
W (i) ([HBn]
In
Hadamard
set of W i l l i a m s o n
= m a x { W ( H n ) ; H n 6 {H~i)}}
W (i) ([Hn])
will
is the
- (i) IH n } , i=2,3
equivalent
.
w(i) (n) damard
set of all
[HB~ i) ] , i=2,3
of o r d e r
of o r d e r
is the
only
and
for n=2
Yang
have
finding
of g e t t i n g
more
difficuld
and n = 0 ( m o d proved
about
a non-trivial one
4).
that
I. W(n) > ( n - l ) ( n + 4 ) / 2 2. W(n) ~ ( n - 2 ) ( n + 6 ) / 2
, for n=47,
t
is an o d d
of v a l u e
number
too.
Note
124
3. W(n)
I + [ n(2n+1)~ ] 2
n(n-1) 2
In 1977 B e s t p r o v e d
2 4. n _
that
( n ) < o(n) < n 3/2
2n
n/2
n 3/2 5. ~
n3/2
~ o(n)
6. ~ (n) = n 3"2 / for H a d a m a r d In
, for n > I
, for s u f f i c i e n t l y
, for a n d o n l y
matrices
[ 94 ] E n o m o t o
for r e g u l a r
w i t h the c o n s t a n t and Miyamoto
large
n
Hadamard
.
matrices,
sum of r o w e l e m e n t s .
h a v e p r o v e d t h a t for l a r g e
I 7. ~([Hn]) ~ n ( ~ )~
In
[127] H a m m e r
a n d all h a d p r o v e d
that
8. ~(n) ~ n 2 ( ( ~ n ) - 2 ) / ( 2 n - 2 n ) 9. ~(22r(
= 23r,
10. o ( 2 2 S ' q 2)
W(22r)
= 22S.q 4
L e t us give k n o w n W(n)
= 23r-1(2r+1) , for
q > 3, s ~ 21og2(q-3)
a n d o(n)
i.e.
for the f o l l o w i n g
n.
n
w(n)
o(n)
n
W(n)
0(n)
2
3
2
36
756
216
4
12
8
40
920
240
8
42
20
44
?
?
12
90
36
48
?
?
16
160
64
52
?
364
20
240
80
56
?
392
24
244
112
60
?
?
28
462
140
64
2304
512
n
hold
125
Let us give
some p r o p e r t i e s
of w e i g h t
and e x c e s s
of H a d a m a r d
mat-
rices:
×
I. o(i) (H I
H 2) =
2. ~(2) (n) =- 0 ( m o d 3. ~(i) (Hn)
~(i)
(HI)O
(i)
(H2)
, i=2,3
4) , n > 2.
= 2w(i) (Hn)-n i, i=2,2
for any H a d a m a r d
matrix
H
n
4. a (i) (-H n) = N i - 2W (i) (-Hn) 5. W (2) (n) ~ 0 ( m o d
2) , n > I
6. W (i) (H n) = n i - 1 ( n + 1 ) / 2 ,
i=2,3
if H
n
is a n o r m a l i z e d
Hadamard
matr ix. 7. W (2) (ran) > m 2 n 2 - n 2 W (2) (m)-m2W (2) (n) +2W (2) (m)W (2) (n) 8. W (2) (n 2) > [n2-W (2) (n) ]2 + [W(2) (n) ]2 9. W (i) (-Hn)
= n i - W (i) (Hn) , i=2,3
H = Qo x I + QI x U +. • "+ Qn-1 x U n-1
10. If
, then
0 (2) (H) = n[o (2) (Qo)+...+0 (2) (Qn_1) ] W
(2)
(H) = n[W (2) (Qo)+...+W(2) (Qn-1) ]
In fact (2) (H) = ~(2) ( n-1 E Qi x U i) = i=0 n-1 E a(2) (Qi x U i) = Z ~(2) (Qi)o(2) (U i) i=0 i=0
n-1
Further,
since o (2) (I) =
so
(2) (U) =...-_0(2) (U n-1 ) = n
t
(2)
By a n a l o g y
n-1 (2) (H) = n Z ~ (Qi) i=0 one can o b t a i n
the v a l i d i t y
of this r e p r e s e n t a t i o n
for
126
W (2) (H). 11.
If the e x c e s s
Hadamard
matrix
H = PO x V O + PI × V1
is of form
+'''+
P n-1 x Vn-I
'
then o(3) (H)
=
n2 n~1
o
(3)
(Pi)
,
i=0 W (3) (H) = n 2 nZIw(3) (Pk) k=0 12. Let us give Note
that
the table
of w(i) (H), o(i) (H),
[HB (i) ] , i=2,3
includes
j
W(2) (Qj)
o(2) (Qj)
0
6
1
i=2,3,
for the m a t r i c e s
H6[HBli)].
Qo,QI,...,Q15,
P0'P1'''"P15 W(3) (pj)
o(3) (pj)
-4
32
0
6
-4
32
0
2
6
-4
32
0
3
6
-4
28
-8
4
6
-4
32
0
5
6
-4
24
-16
6
6
-4
28
-8
7
6
-4
24
-16
8
10
4
40
16
9
10
4
36
8
10
10
4
36
8
11
10
4
32
0
12
10
4
36
8
13
I0
4
32
0
14
10
4
32
0
15
10
4
28
-8
127
13. ~(2) ([HB4t]) -= 0(rood 4t),
0 (3) ([HB4t]) =-0(rood 4t 2)
The p r o o f
f r o m items
of i t e m
13 f o l l o w s
10
12
14. -4t2 < 0 (2) ([HB4t]) < 4t 2 , 6t2 < W (2) ([HB4t]) < 10t 2 15. - 1 6 t 3 < 0 (3) ([HB4t]) _< 16t 3, 16. n(n-1) (n+4)
24t 3 < W (3) ([HB4t]) < 40t 3
< W(3) (n)< n2(n-1) --
+ n[n(2n+1)I/2
--
]
2
2
2
n32-n,n ) _ (3) (n) < n 5/2 ~n/2 < o P where
p
means
t h a t we c o n s i d e r
only three-dimensional
regular
Hada-
of r e g u l a r
Hada-
mard matrices. The p r o o f
of i t e m
mard matrices,
theorem
DEFINITION of H a d a m a r d
16 f o l l o w s
6.6.
matrix
PW (i) (H n)
:
1.3 of
[ 7 ]. A Hn
of H a d a m a r d
~ 7 ] a n d l e m m a of
(maximal)
W (i) (H n)
(PW
(i)
matrix
H
n
of o r d e r
(n) -
n
of o r d e r
n
Note that there sity a n d e x c e s s
of H a d a m a r d
STATEMENT
6.4.
511
Let H
(maximal
+
relation
i
between
po(i)
den-
(Hn)]
be a n o r m a l i z e d
Hadamard
m a t r i x of o r d e r n.
Then
= I-(I 2
the w e i g h t
matrices
n
pW(2) (Hn)
Po (i) (n))
(n)) n
= I
)
(i)
i
pw(i) (H n)
is the r e l a t i o n )
is the r e l a t i o n
is the f o l l o w i n g
density
i=2,3
i
, (Pc (i) (n) = q n
pw(i) (Hn),
Po (i) (Hn) , i=2,3
o(i) (Hn) Po (i) (Hn)
n
W (i) (n)
i
6.7.[ 7 ]. A d e n s i t y
[215].
weight density
(excess d e n s i t y
n DEFINITION
f r o m the d e f i n i t i o n
+ I ~) ; P
(2)
(Hn)
I = n .
128
STATEMENT
6.5.
It is true
3/8 < pw(i) ([HBn ]) < 5/8,
i=2,3
-1/4 < Pa(i) ([HBn ]) < I/4, i=2,3
The proof follows STATEMENT I. lira
6.6.
from items
14 and 15.
It is true
PW (i) (n) = I/2,
i=2,3
n--~ 2. lim
Pa (i) (n) = 0, i=2,3
n-~=
6.3. C o n s t r u c t i o n
of t h r e e - d i m e n s i o n a l 9eneralized
The classes of so called
tion
of a b o v e - m e n t i o n e d
Hadamard matrices
using the algebraical [239]
were c o n s t r u c t e d
apparatus
he overcame
Hadamard matrices
spatial generalized
which are the g e n e r a l i z a t i o n generalized
(high-dimensional)
Hadamard matrices hogh-dimensional
by Egiasarian
of high-dimensional
the difficulties
and
C.O.
matrix multiplica-
in desoription
of different
classes of special Hadamard matrices. The main problem mard matrix
is the construction
of spatial generalized
Hada-
[H(p,m) ]n for natural numbers p, m, n.
We will use a b o v e - m e n t i o n e d d e f i n i t i o n of
(l,~t)- orthogonal
algebraical
apparatus
spatial matrix
for the general
[ 239].
Let us denote by
[A]n
=
II Ail,i 2 ..... in
II
;
[B] r
=
II B.
31,j 2 ..... jrll
,
(il,i 2 ..... in,Jl,j 2 .... ,Jr=l,2 .... ,m)
n-dimensional
and r-dimensional
(l,~)-convolute
matrices of order
product [239] of matrix
[A] n
m
by [B] r
respectively. over the parti-
129
tion
incides
s
and
c
[D] t = II Dl,s,kll
where
will
a matrix
[D] t
provided
= l'g([A]n,[B] r ) =If Zc A 1 ,s,c B c , s , k II
n= x+l+~t , r=v+%,+~,
numbers,
be c a l l e d
i=(11,12,...,Ix) , x,l,b,o
s=(sl,s2,...,s)),
c=(cl,c2,...,c
(6.14)
- non-negative
) , k=(k 1,k2,...,k
),
of o r d e r
m
be a con-
be t r a n s p o s e d
H'
t=n+r-l-2~ • L e t n o w H' b e a n - d i m e n s i o n a l jugate
to H' m a t r i x
ces respectively,
matrix
a n d H ' t a n d H''t
over
the
definite
indices
, H''
a n d H''
matri-
(t is a f i x e d n a t u r a l
num-
ber) . DEFINITION called
a
fied the
6.8.
(l,~)-orthogonal
k=n-l-5,
(H[H[') E(l,k)
in all n o r m a l
= m~E(l,k)
is a
a)
6.3.
The notation
those
Hadamard
the
following
X = 0, matrix.
~ = n-1 The
, for
of
c a n be
satis-
6.15)
unit matrix
and
~ ~ k
, for
~ = k
(l,~L)-orthogonal
Hadamard
of H t'- m a t r i x
spatial
matrix
colnci-
matrix
are p-th
[H(p,m)]n,
roots
of u n i t y .
for Let
cases. we have
system
0,n-1(HiHi,)
where
will be
, t=1,2,...,N
generalized
if the e l e m e n t s
us consider - for
directions
m
of
three-dimensional
~+~ = n - i
axis
of order
{ n!/2X!~!k!
des with
H' t
(l+2k)-dimensional
n!/l!~!k!
N= NOTE
matrix
conditions
t, ~
where
A n-dimensional
(6.15)
(general)
n-dimensional
generalized
becomes
= mn-IE(0,1)
(6.16)
130
H t' = H'
" ' " ~t) ''" ±I
(~I i2 12 13
(~t it+1 , H~'
=
H"
in
"'" in
it
"'"
in-1
t=1,2,...,n
- for I= n-2 generalized n(n-1)/2
, ~ = I
Hadamard
equations
we h a v e c o m p l e t e l y
matrix
if s a t i s f i e s
obtained
n-2'l(Htl,t
from
n-dimensional
the f o l l o w i n g
s y s t e m of
(6.15).
H"
2
proper
)
tt,t 2
=
mE(n-2
'
(6.17)
I)
where (
H' =
H !
tl,t 2
i I i 2 "'" iti-I
it
i2 i3
t11
..- iti
i I i 2 ... i t it (i 2 L 3 i I -I i11 = H" "'" tl
' ' it21t2+1 in it 2
... i n ... in_ I
it i + ... i n in2 it2 1 i ) t2 n-1
H ~
t I ,t 2
t1=1,2,...,n-1
Note
that
equation b) n-l,
for n=2
the
system
f r o m the d e f i n i t i o n
spatial
(special)
a n d if t a k e s p l a c e
, t2=1,2,...,n
(6.15)
coincides
of g e n e r a l i z e d
orthogonal
matrix
the o r t h o g o n a l i t y
w i t h the k n o w n m a t r i x
Hadamard if in
over
matrix.
(6.15)
k=2,3,°..,
set of d i r e c t i o n s
12,
i=1,2,...,n. NOTE
6.4.
it s a t i s f i e s
If the for
system
I =11~o)
Let us g i v e a r e c u r r e n t generalized
(6.15)
Hadamard
satisfies
for I =Io(~=~o) , t h e n
too.
m e t h o d of c o n s t r u c t i o n
matrix
[H] n = II h(n) II il,i2,-..,i n
[H(p,m) ]n
of o r d e r
of n - d i m e n s i o n a l m
,
, il,i 2 ..... in=0,1 .... m-1
131
from
the
generalized
H (p,m)
Hadamard
II
[H] 2 =
Yp h e r e a) order
=
and after Suppose m
II ={ hit,12
denotes
that
matrix
(ii 'i2) }m-1 ii,i2= 0
Yp~
the o r i g i n a l
the k,dimensional
p-th
root
of u n i t y .
generalized
Hadamard
matrix
of
is c o n s t r u c t e d :
[HI k :
we c o n s t r u c t
II h!31,J k) 2 .... , jk 1I
the m a t r i x
=0,1,2,...,mi-I
[A] k =
obtained
from
, j1,j 2 ..... J k = 0 , 1 , 2 ..... m-1
II a(k) L , 11,12,...,in = 11,12,...,i k d i r e c t p r o d u c t of m a t r i x [H] k into
itself. Then
[A] k =
II a(k) m i 1 + J l ....
,mik+ik iI
=
(k) (k) II hll , .... ik " h 31 ' ' ... 'Jk
II (6.18)
il,i2,...,ik,Jl,J2,...,Jk=0,1,2,...,m-1 b)
L e t us d e f i n e
a
(k+1)-dimensional
.
[H]k+ I =
matrix
of o r d e r
m.
a (k)
h (11 k +,i 1 )2 , • . . ilk+ I II =
II
(m+1)i I , (m+1)i2. • . ( m + 1 ) i k _ I ,
ik÷ 1 11
which
is the
Having sional
(6.19)
spatial
the m a t r i x
generalized
generalized
Hadamard
[HI 2 , (6.18)
Hadamard
and
matrix. (6.19)
we o b t a i n
the
n-dimen-
matrix.
n-12n-l- I
[~]n =
Let us n o w g i v e
BIYp
• ~(ii,i2)
+ ~ ( i l , i n)
1=2
an a l g o r i t h m
Ji
for the c o n s t r u c t i o n
of c o m p l e t e l y
pro-
132
per
spatial
Hadamard
[B] 2
be a g e n e r a l i z e d Vandermonde
matrices
[H(p,p) ]n. Let
p-1
: II b!2) II = { ypil,i2} zl,i 2
Hadamard
matrix
matrix
[60].
...,in=0,1,...,p-1
H(p,p)
The matrix
, we define
il,i2 =0
constructed
according
[B]
=If b!n) . n 11,...,l n by the r e c u r r e n t m e t h o d
II b(n) II = II b(n-1) . il,i2,.-.,i n i1+in,i2+in,i3,--.,in_1
to
, ii,i2,...
II
, n>2
(6.21)
or
(2) " II b!n) 11 ..... in II = II b i I +i3+ . .+in,i2+i3+ . . .
=If y p ( i 1 + i 3 + ' " + i n )
One can verify ly p r o p e r Give
spatial
that
the matrices
generalized
an example
lized H a d a m a r d
(i2+i3+'''+in)
of c o n s t r u c t i o n
matrix
H2 =
II Bil,i2,i311 =
(6.22)
n=2,3,...,
matrices
are c o m p l e t e -
of type
of c o m p l e t e l y
proper
[H(p,p) ]n. cubic
[H(3,3) ] 3. Let II
be a g e n e r a l i z e d
=
II
[B]n,
Hadamard
+inI[
H(3,3)
I
I
I I
I
I
I
XI X2
I
X2 X I
Hadamard
I
I
matrix.
Then B=II B . . . II 11,12,13
X I X2
XI I
X2
~
(i I )
X 1X 2
X2 X 1 1
I
1
~
(i3)
X2 XI
I
X2 1
I
I
I
XI
(i2)
ii,i2,i3=0,I,2
genera-
133
is the completely
proper
cubic generalized
Hadamard
matrix
[H(3,3) ] 3.
Chapter
3. A P P L I C A T I O N OF H A D ~ A R D
MATRICES
The m a i n r e s u l t s of first two c h a p t e r s have for d i f f e r e n t b r a n c h e s of m a t h e m a t i c a l We w i l l give
several applications
and e n g i n e e r i n g c y b e r n e t i c s .
some of these a p p l i c a t i o n s
for i n f o r m a t i o n theory,
const-
r u c t i o n t h e o r y etc.
§ 7. H a d a m a r d m a t r i c e s and p r o b l e m s of i n f o r m a t i o n theory
7~I. H a d a m a r d m a t r i c e s and b i n a r y codes.
Let us give the defi-
n i t i o n of a code. DEFINITION n
7.1.
(with c o m p o n e n t s
d i f f e r at least
in
[157 ].
(n,M,d)-code
f r o m some d
is a set M of v e c t o r s of
length
field F9 such that e v e r y two v e c t o r s
p o s i t i o n s and
d
is the g r e a t e s t n u m b e r w i h h
this property. We w i l l c o n s i d e r
the b i n a r y codes that
Let us denote by M=M(n,d) ry
is c o d e s
the g r e a t e s t n u m b e r
for w h i c h F={0,1}
of code w o r d s
in e v e -
(n,M,d)-code. Note
that
in g e o m e t r i c a l
sence
the m a i n p r o b l e m of c o d i n g t h e o r y
is the c h o i c e of p o s s i b l e g r e a t n u m b e r of v e r t i c e s of a cube w i t h a given upper estimate pairwise
distance
( n , M , d ) - c o d e m e a n s the c o n s t r u c t i o n of of r a d i u s
d/2 w i t h c e n t r e s
and the c o n s t r u c t i o n of the M
non-interesting
in v e r t i c e s of a cube,
i.e.
spheres
this p r o b l e m
is the p r o b l e m of packing. Bose and S h r i k h a n d e
(1959), Mc W i l l i a m s
and Sloane
p r o v e d that H a d a m a r d m a t r i c e s a l l o w to c o n s t r u c t
(1979)
have
the f o l l o w i n g
four
codes. T H E O R E M 7.1.
If there e x i s t s the H a d a m a r d m a t r i x H
n
of o r d e r
n
t h e n there e x i s t s I. the
(n-l,n,n/2)-code
(consisting of rows of m a t r i x H
n
without
135
first
column);
2. t h e de a n d
(n-1,2n,n/2-1)-code
their
3. the
(consisting
of v e c t o r s
of p r e v i o u s
co-
complements);
(n,2n,n/2)-code
(consisting
of r o w s
of m a t r i x
H
n
and
their
complements). 4. the Note and
that
secondly, Using
one
(n-2,n/2,n/2)-code. firstly, all
above-mentioned
the m e t h o d s
can construct THEOREM
there
7.2.
exist
(n-l,n,n/2)-code
Let H
codes
with
gers
i= P ~ o and
"P~o
following
(nl-1,
,
2nl,
,
~ i ~ 0,
1961Plotkin
a)
if
d
stated
is e v e n
[1881 from
§ 4
is t r u e .
of o r d e r
n
. Then
parameters:
ni/2-I)
,
,
i=0,I ..... k,
are
arbitrary
inte-
that
then
2[d/(2d-n} ] M(n,d) <
b) if
d
(7.1)
for
n = 2d
(7.2)
(7.1),
7.3.
matrices
2[ ( d + 1 ) / ( 2 d + 1 - n ) ] , for d < n < 2d+I
(7.3)
2(n+I)
(7.4)
< {
Levenstein
the r e l a t i o n s THEOREM
d < n < 2d
is o d d t h e n
M(n,d)
1964
, for
{
2n ,
damard
matrices
that
matrix
for n=2 k
[188]
Pi/26{1,2,...,50}.
In
In
ones,
(nl-2,nl/2,nl/2)
"...'P~k
linear
of H a d a m a r d
from given
the
,
(nl,2nl,nl/2)
are
be an Hadamard
n
(nl-l,nl,nl/2)
where
codes
of c o n s t r u c t i o n
new codes
is the e q u i d i s t a n t - c o d e
has (7.2),
,
found
for n = 2d+I
the n e c e s s a r y
(7.3),
(Levenstein,1964). then
on P l o t k i n
(7.4)
conditions
converting
to the e q u a l i t y .
If t h e r e
boundaries
exist
(7.1)
-
corresponding (7.4)
Ha-
equalities
136
are hold
that
Codes
is there e x i s t
on P l o t k i n
and s o m e t i m e s
boundaries
optimal
In a b o v e - m e n t i o n e d for a g i v e n n)
n
M(n,d)-2n
paper
these b o u n d a r i e s .
are c a l l e d m a x i m a l
a method
codes
of c o n s t r u c t i o n
that there e x is t
It also p r o v e d
holds
reaching
[163,
237].
codes.
(provided
is proposed.
codes
if and only
there
if
that
Hadamard
matrices
for d=2k,
4k is the order
of m a x i m a l
code
of o r d e r
n=2d the e q u a l i t y
of some H a d a m a r d
mat-
rix. Note linear
that the m a x i m a l
for M=2-
are used here) problems codes
codes
constructed
and n o n - l i n e a r
for d2-d+4 > 4t
and that this m e t h o d
dictate
on P l o t k i n
construction boumdary
by L e v e n s t e i n
so as to s i m p l y t y
are
(Paley c o n s t r u c t i o n s
is a direct
(by p o s s i b l e
method
one.
simple
Some p r a c t i c a l
method)
the m e t h o d
of linear
of code c o n s t -
ruction. I think ly us i n g maximal
that u s i n g
the r e c u r r e n t
parameters
des
structure
method)
of H a d a m a r d
matrices
one can c o n s t r u c t
more
(especial-
simple
linear
codes.
Let us d e m o n s t r a t e
order
the
this by a simple
(2km,2km,2k-lm)
of some H a d a m a r d m2+2m
and
and
matrix.
m 2 bits,
example
of maximal
(2km-1,2km,2k-2m), We will use
respectively.
where
m
for g e n e r a t i o n
Introduce
codes w i t h is the of these co-
the f o l l o w i n g
nota-
tions. Am
is a
(0,1)-matrix
first r o w of this ce by Ei m
obtained
(0,1)-matrix
from H a d a m a r d
consists
matrices
completely
of
of order
m;
I and -I repla-
0; is a c o l u m n - v e c t o r
of length
Am = J-
Am
Am = II E°A il
m
coordinates
of which
are
i;
137
Eo
2k-lm
k-1 m
2k-lm
2k-lm
Eo 2k-1 m
k-1 m
A' 2k-I m
A' 2k-1 m
A 2km
A
!
EO 2k-lm
A' 2k-lm
A' 2k-lm
EI 2k-lm
A' 2k-lm
A' 2k-lm
=
2km
It can be easily noted that the matrices above-mentioned
maximal
In conclusion I)
codes.
was used in telemetric
de which Read-Maller
from the Hadamard matrix of order
to construct
new optimal
7.2. Hadamard matrices Let CI,C2,...,C T
of Hadamard matrices
code
7.2.
N=2 k coinci-
and multiple-access
channels.
be bimary codes of length
[328]. Let
(see 4) allows
codes from given optimal codes.
these codes is a T-user code of length
n.
69
codes of first order;
3) new method of construction
length
system Mariner
1968);
2) codes o b t a i n e d
DEFINITION
form the
let us note that
(32,64,16)-code (Posner,
A2km, A'2km
n
n. The totality of
denoted by
(C1,C2,...CT).
(C1,C2,...,C T) be a T-user code of
(CI,C2,...,C T) will be called uniquely decodable
if for every vectors
(codewords)
Ui,Vi 6 Ci,
I< i
T-user
the condi-
tion
T xui# i=I
T zv i=I l
is true. The p r o b l e m of c o n s t r u c t i o n
of uniquely decodable
codes was in-
138
vestigated (1971),
for
Slepian
and Weldon T
first
(1973),
(1979),
binary
a common find
the
by
Kasami
codes
(1961) (1978),
and
later
Weldon
by M e u l e n
(1978),
Chang
(1981,1983).
C I , C 2 , . ° . , C T are
preserving
CI,C2,...,C T
Shannon
and Lin
Khachatrian
codes
channel
time
bit
transmitted
simultaneously
synchronization.
of m i n i m a l
length
n
The
problem
allowing
the
over
is to uniquely
decoding. If we a l s o
suppose
ICi [= I,
I < i <~
decodable
basic
Let x = ...,T}
that
, then
of
( C i , C 2 , . . . , C T)
Ci
will
code
is 2 t h a t
be c a l l e d
is
T-user
uniquely
code.
( X l , X 2 , . . . , x n)
is the
the p o w e r
L-distance
and y =
between
( y l , Y 2 , . . . , y n)
vectors
x
where
and
y
x1,Y16{0,1,
defined
as
n
dL(X,y)
DEFINITION if for e v e r y Ci
the
7.3.
[328].
vectors
following
= E Ix i - yil i-0 (CI,C2,...,CT)
(UI,U2,...,UT)
and
is a
6 -decodable
( V I , V 2 , . . . , V T) w i t h
code Ui,Vi6
condition
dL(U I + U 2 +...+
UT,
V I + V 2 +...+
V T)
h 6
the
following
holds. From
Then
above-mentioned
THEOREM
7.4.
there
exists
Let
Let
papers
there
exists
the T - u s e r
(CI,C2,...,CT)
one can the
basic
be a T - u s e r
obtain
Hadamard
n-decodable code
of
matrix code
length
theorem.
of o r d e r
of
length
n
.
n
n.
log21cil Ri -
is c a l l e d
a rate
of c o m p o n e n t
Ci,
Rt=RI+R2+...+RT
n
is c a l l e d
a common
rate
Let us n o w c o n s i d e r
of
(Ci,C2,...,CT)-Code.
uniquely
decodable
basic
codes;
R.
= I/n,
l
T t = T/n. Let
( C I , C 2 , . . . , C T) be a b a s i c
code
that
is
ICi[
= 2, C i = { U i , V ~ .
t39
Vector
di = Ui - Vi
C i and matrix
is c a l l e d
a difference
D = [ d l d 2 . . . d T IT is c a l l e d
vector
of the c o m p o n e n t
a difference
matrix
of
(C I,
C2,...,CT)-Code. If t h e r e C i are
is a d i f f e r e n c e
defined
d =•
as
follows.
i ' °i =
U0
matrix
D
then
L e t us
introduce
vi=
Inj=1'
I
2 codewords the
following
of c o n p o n e n t notations.
i=l,2,...,T
0 for dO = 0 or dO = -I l l
=
i
(7.5)
I for dO = I l
0 for dO = O o r d O Vj~ = l
:
1
l
I for d~ = -I 1 THEOREM code
7,5,
if a n d o n l y
[322].
(CI,C2,...,CT)
if for e a c h
is an u n i q u e l y
(0,-1,+1)-vector
M the
decodable
basic
condition
MD ~ 0 holds
(MD = 0
Thus, sic c o d e
only
the p r o b l e m of
length
(0,-1,+1)-matrix arly
if M = 0) .
n D
of c o n s t r u c t i o n is r e d u c e d with
of T - u s e r
to the
dimension
T × n
uniquely
decodable
problem
of c o n s t r u c t i o n
the
of w h i c h
rows
are
ba-
of line-
independent. In
[ 32~
ven.
The
ted,
where
and
n
two methods
difference Tk =
for c o d e s
of c o n s t r u c t i o n
matrix
with
(k+2)2 k-l,
dimension
nk=2k
constructed
of d i f f e r e n c e Tk × nk
for e v e r y
are
also
k.
matrices
is t h e r e
construc-
The p a r a m e t e r s
T
given
n
I
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
T
I
3
5
8
10
13
15
20
22
25
27
32
34
37
39
48
THEOREM
7.6.
If t h e r e
exists
are gi-
a T-user
uniquely
decodable
basic
140
code le
of
length
basic
code
PROOF. basic
of
Let
code
n
of
, then length
D o be
nl I
a
length
there
1=I
t
t
Let
exists
2,
difference n.
Di =
also
Tll-user
of
consider
(X + Y) × D i = I
decodab-
.o.
matrix
us
uniquely
T-user
the
uniquely
decodable
matrix
, i=I,2,...
(7.6)
where 1
I
1
1 -1
I
I
I -I
1
1
1 -1
-1
I -I
-1
1
X + Y =
(7.7) I -I
We T'61
will
× n6 !
every
prove
non-zero
-I
I
I I
-I
I -I
I -I
-I
I
I
that
. According
-I
D. 1 to
I
is
the
the
I -I
difference
theorem
(0,-1,+1)-vector
7.5
V
of
matrix
we
need
length
6n
with to
dimension
prove
the
that
for
inequality
VD I ~ 0 holds. Represent -th
vector
portion
M
=
{ (X I + X 2 -
=
of
V
in
vector
form V
= VD I = V(X
of
+ Y)
V
=
(XI,X2,...,X6),
length
Do
n
. Let
us
where
Xi
is
the
calculate
=
+ X3
+ X4
- X 5 - X6)Do,
(X I + X 2
+ X3
- X4
+ X5 -
+ X3
- X4
- X5
+ X6)D O,
(X I - X 2
- X3
- X4
+ X5
- X3
+ X4
- X5
+ X 6 ) D O,
(-X 1 - X 2
X6) D O , (X I + X 2
+
+ X6)D O ,
=
(-X I + X 2 -
X6)
+ X3
+ X4
+ X5
-
O O}
(7.8)
141
It
is k n o w n
ve
that
M
tement, From
that
X.D i o
~ 0 for
every
i.e,
(7.8)
let
~ 0
and
X.D 1 o
non-zero
there
exists
= 0
if
vector
V
a non-zero
and
only
. Suppose
vector
V
if the
such
X i = 0.
Pro-
reverse
sta-
that
M
= 0.
follows
(X I + X 2 + X 3 + X 4 - X 5 - X 6 ) D °
= 0
(X I + X 2 + X 3 - X 4 + X 5 - X 6 ) D °
= 0
(X 1 + X 2 ÷ X 3 - X 4 - X 5 + X 6 ) D O
= 0 (7.9)
(
-X 1
(-Xl
+
can
-
the
proved
COROLLARY Let
+
system
contradicts be
X 3
X 4
- X2 + X3 + X4
Solving what
X 2
us
to
-
(7.9)
give
for
condition
The
the
°
So
of
=
the
the
= 0
obtain
theorem.
theorem
rate
of
D
n
1=10
i=1,2,...,6
the
theorem
and
o
and
T
length
n
D. is e q u a l . i of
constructed
n
2
4
6
8
10
12
14
16
20
24
28
32
40
48
56
64
T
3
8
13
20
23
32
33
48
56
76
80
112
132
172
188
256
96
T
304
400
7.3.
We
Had amard
will
prove
matrices
here
and
that
F
(where
H
is
tructed
in
§ 4 Hadamard
are
solutions
the
an
Hadamard
of
data
the
processing
orthogonal
Hadamard
transformations
= HE
matrix matrices
such
of
For
: 0,
uniquely
00
codes
X D i o
is p r o v e d .
codes
parameters
basic
0
T-user
n
decodable
X6)D
X.D we i O
of
common
table
+
+ X 5 - X6)D °
analogously. 7.1.
X 5
problems
(7.10)
and
f
is a n
(of c o u r s e , in d a t a
on
input
vector)
certain
processing
for
cons-
conditions) as
effective
142
and n o i s e l e s s coding,
filtration,
pattern recognition
(mathematical
f o r m u l a t i o n of the p r o b l e m will be given later), with that and in view let us note that in several t h e o r e t i c a l and a p p l i e d p r o b l e m s an Important role plays r e p r e s e n t a t i o n of the functions as a sum of more simple o r t h o g o n a l components.
Besides,
definite o p e r a t i o n s with functions
come the o p e r a t i o n s with spectral components. ral a n a l y s i s
Of p a r t i c u l a r
is spect-
for m i n i m i z a t i o n of some criterion.
Let us give the common formulation of a p r o b l e m of choice of optimal base in f i n i t e - d i m e n s i o n a l F o r m u l a t i o n of prob!em.
Let
space R N. x c R N be a set of initial functi-
ons, ~ a be a set of bases R N satisfying the definite conditions, S(xi,~ a) be a functional defined on set of initial functions of ~ and
p a r a m e t e r of
S
(bases
a). The p r o b l e m is to find a base ~o, minimi-
zing the value S(xi, ~ ) :
S(xi ~O)
Coordinate
system
= min a
min S(xi~ a) ,$6,~
(7.11)
C ° said to be optimal by c r i t e r i o n S system.
Give some formulations of this type. I. Best a p p r o x i m a t i o n space R N with norm product, {~i
=
{~i }N i=I
}N
Ci i i=I
in Hilbert space.
Let
p(x,x #) = , where
x
be an E u c l i d e a n
<x,x#>
is some fixed o r t h o g o n a l base,
is the scalar
~ is a set of type
N systems,
C i are constants.
S(x;~)
=p
(x-
E ~i ) . The i=I
best base is
{<
2. D e c o m p o s i t i o n
x,~i> ~i }Ni=I
into singular values
space of m a t r i c e s of order {~i }
and{,p~}
N xN
(rank of the m a t r i c e s
be two o r t h o g o n a l bases
ak) be a m a t r i x of order
(DSV) [19J]. Let
x
be an
is N). Let
in RN-space and Gk(al,a2,...,
N x N, such that
143
Gk =
S(G;{~};{~i};
According of G - m a t r i x values
to t h i s
with
,
a l , a 2 ..... a k)
criterion
parameters
of G - m a t r i x .
k Z ai~i~oit i=I
the b e s t
a i , whose
Moreover,
=
the v a l u e
II G - Gki I
base
in the
values
(7.12)
2
singular
are e q u a l
of c r i t e r i o n
S
base
to s i n g u l a r is
N
IIG-
%112
=
z i=k+1
3. K a r h u n e n - L o e v e mensional RN
, k
vectors,
be
some
decomposition x
be a r a n d o m
positive
denotes
the
genvectors
subspace
[1941. Let subspace
R N be a space
of R N , { ~ i }
of N - d i -
be a b a s e
in
integer.
S ( ' ; { ~ i} : E ( p ( f
E
~2l
k - I < f, ~ i > ~ i i=I
average.
~i of m a t r i x
G(i,j)
Minimum
(7.13)
))
of
S
= E(f(i) "f(j)),
gives
the base
of ei-
and
N
2 S(" ;{~i }) : i = ~ + 1 1 i where
ii
Note
are e i g e n v a l u e s that
of m a t r i x
eigenvectors
G.
of H e r m i t i a n
matrix
A
maximize
Relay
re-
lation
p(x)
4. M i n i m i z a t i o n dom
subspace
of e n t r o p y
RN with p(x)
=
= < A(x) ,x> < xtx >
of a r a n d o m
,
scurce.
(7.14)
Let
x
be a ran-
density-probability function I I e2 (x-mi) TC-I (x-mi) n
I
(7.15)
144
{~i }
be an
o r t h o g o n a l base,
Yi =< x'~i>
' S(';~)
= -~p(Y) in p(Y)dy
The base m i n i m i z i n g e n t r o p y of a r a n d o m source with d e n s i t y - o f - p r o b a b i l i t y function p(x) Obove-mentioned theoretical
is the base of e i g e n v e c t o r s of matrix
C .
formulations of p r o b l e m s are of the fundamental
importance and find widely p r a c t i c a l applications.
decomposition
So,
into singular values and K a r h u n e n - L o e v e d e c o m p o s i t i o n
are used for p r o c e s s i n g of digital signals m a x i m i z i n g Relay r e l a t i o n a d e q u a t e to p r o b l e m of filtration of legitimate [323].
In these p r o b l e m s optimal by c r i t e r i o n
g e n v a l u e s of some symmetric matrix.
signal from the noise S
is the base of ei-
It was m e n t i o n e d in general formu-
lation of p r o b l e m that the base r e q u i r e d is looking for between the bases s a t i s f y i n g the definite conditions,
and a l t h o u g h f o r m a l i z a t i o n
of these c o n d i t i o n s and their c o n s i d e r a t i o n are sometimes
impossible
they play n e v e r t h e l e s s an important role in choice of p r a c t i c a l l y r e a l i z a b l e base.
In particular,
these c o n d i t i o n s are often as follows:
a) number of zeros and number of e x t r e m a of base functions on finite interval
("oscillation",
"frequency")
are m o n o t o n i c a l l y
increasing
w i t h the number; b) c o n v e n t i o n a l
in physics and technics functions are a p p r o x i m a b -
le p r e c i s e l y e n o u g h with the finite and not very large number of decomp o s i t i o n e l e m e n t s over these systems of functions
("contains not ve-
ry large number of harmonics"); c) c o r r e s p o n d i n g discrete o r t h o g o n a l simple e n o u g h d i s c r e t i z a t i o n . n
systems are o b t a i n a b l e by
The set of integers n , for w h i c h first
d i s c r e t e o r t h o g o n a l functions are the complete base
in
n-space,
is large enough; d) d e c o m p o s i t i o n c o e f f i c i e n t s p r o v i d i n g p r e c i s i o n m e n t i o n e d can be "fastly" c a l c u l a t e d
(effectiveness of analysis),
be e f f e c t i v e l y r e s t o r e d by these c o e f f i c i e n t s
the function can
(effectiveness of synt-
hesis); e) the system functions are g e n e r a t e d by h a r d w a r e e f f e c t i v e l y and
145
s imp ly. Items b) and c) are
so c a l l e d
is the b a s i s of p o t e n t i a l on training. et al
1964.
broken" rema
These
items
we will
By "common" Lipshits precesion
He(L),
of best
formulation
c) p o l y n o m i a l ximate
the
of c)
The c o n d i t i o n s tions
in practice.
wing.
Let us have
composition
of
from H follows
Note
that by
=
from known
%0m(k) are o r t h o g o n a l can be r e w r i t t e n
a = Cf
of ext-
function.
the f u n c t i o n s
class.
the
of the
from
of a p p r o x i m a t i o n by t r i g o n o m e t r i c Thus,
the m a t h e -
system must
appro-
O(n-2) .
Kotelnikov
theorem
for a p p l i c a t i o n
formulation
function
f(n)o
of item d)
[35]. of d e c o m p o s i -
is the
follo-
The c o e f f i c i e n t s
of de-
form
n ~ f(n)~m(k) k=1
of the f u n c t i o n
=
number
for this
provided
functions
(L) to w i t h i n
Mathematical
have
"not very
zero of d i s c o n t i n u o u s
degree
over He(L)
by A i z e r m a n
is n
a tabulated
large
to u n d e r s t a n d
from d) are n e c e s s a r y
f(k)
formations
"not very
the p r e c i s i o n
item c)
a(m)
where
functions
which
of a u t o m a t i -
in form d i s c u s s e d
0 < ~_< I , by s u f f i c i e n t
approximation
mentioned
Restoration
having
it is natural
from first
function
Necessity
in the p r o b l e m
imply by c o m m o n
functions.
we will u n d e r s t a n d
polynomials matical
they
here
of c o m p a c t n e s s "
imply the p o i n t of sign change
functions
class
method
are given
fanciful",
in a small domain"
function
function
In this paper
or "not very
"hypothesis
is p e r f o r m e d
by f o r m u l a s
n I a(m)~m(k) m= I functions.
(7.16)
The pair of these
in the m a t r i x
,
f = ~Ta
Fourier
trans-
form
(7.17)
146
where tively,
f
and
a
are vectors of f u n c t i o n and of c o e f f i c i e n t s respec-
F is the m a t r i x with elemrnts ~i(j) . The c a l c u l a t i o n of decom-
p o s i t i o n c o e f f i c i e n t s and the r e s t o r a t i o n of function by demands p e r f o r m a n c e of and synthesis consists of t r a n s f o r m a t i o n s
O(n 2) operations.
a
f
E f f e c t i v e n e s s of a n a l y s i s
in c o n s t r u c t i o n of a l g o r i t h m s for c o n d u c t i o n
(7.16),
(7.17) d e m a n d i n g
O(n log) 2n) operations.
The most i n v e s t i g a t e d functions having wide a p p l i c a t i o n s ce are t r i g o n o m e t r i c
and
functions,
Lipshits,
in practi-
H e r m i t i a n functions,
on. R e c e n t l y much a t t e n t i o n is given to i n v e s t i g a t i o n Haar,
and so
(both t h e o r e t i -
cal and practical)
of W a l s h - H a d a m a r d ,
Slant functions
[24] Walsh-
H a d a m a r d functions
satisfy the c o n d i t i o n s a) - e) and relative to condi-
tions for b i n a r y data p r o c e s s i n g and for 2 - d i m e n s i o n a l data input in computer. Let us show that for system c o n s t r u c t e d b e l o w the c o n d i t i o n s a)e) are hold.
Define a system of p i e c e w i s e c o n s t a n t functions after the
example of sequence of e m b e d d e d H a d a m a r d m a t r i c e s
H
from § 4. 2n k
j had2nk+1 (x) = [H2nk+ I] i,j
'
j+1
x6[2nk+1
i,j = 0,1,...,2n k+1 - I,
2nk+1]
k=0,I,2,...
Note that all the system is formed from subsystems and the maximal number of function signs changes from
k-th subsystems doesn't exceed
2n k . H a d a m a r d functions are orthogonal on [0,1]
I
, that is
I, if k=m and i=j
] had k (x)had k 0 2n +i 2n +j
(x)dx ={ 0, if k~m or i#j
a) by frequency of H a d a m a r d function we will u n d e r s t a n d the number of sign changes on
[0,1]
("rate of changes").
This n o t a t i o n was given
147
for W a l s h
functions
of f u n c t i o n
in the
(the
special
system
case
doesn't
of
H
decrease
system) The f r e q u e n c y 2n k f r o m one s u b s y s t e m to a n o t -
her. b) bered
let
f(x)
functions
be
integrable
of the
function
system.
Let
on
[0,1],hadn(X)
us c o m p o s e
the
be a n e w n u m -
series
oo
E C h a d (x) n n n=1
where I
C
=
~f(t) 0
n Sm(f;x)
denotes
had
(t)dt n
a partial
sum of the
m
Sm(f;x)
From
series:
m
= E Cnhadn(X) n=1
orthogonality
: Sf(y) [ E h a d (x)hadn(Y) ] dy n=1 n
of H a d a m a r d
S
follows
immediately
that
= I__ S f ( y ) d y
(f;x 2m
2m d m
where
x 66 m
,_ i t J _ 2m
=[ 2 m
] ,
Hence, _
f(x)
- S2m(f;x )
I
2m
I[f(x)
- f(y)]
dy
m
We o b t a i n THEOREM
I. S
immeadiately
7.7.
(f,x) 2n k
Let
the
the v a l i d i t y function
converges
to f(x)
f(x)
of the be
following
integrable
in a l m o s t
on
all p o i n t s
theorem. [0,1].
of
Then:
(0 I);
148
2. in point
X O of c o n t i n u i t y
f(x),
S
(f;x)
converges
to f(X O)
2n k 3. if f(x)
is c o n t i n u o u s
on
(0,1)
then
S
(f;x)
converges
to
2n k f(x)
uniformly
Denote by H THEOREM
(L)
7.8.
in
X.
the class of L i p s h i t s
If f(x) 6 H
If(x)
-
S2nk(
(L)
functions.
then
(I_!_)
L
f;x) Ij
~+I
2n k
for all
x 6 [0,1]
The proof
of this
, k=0,I,2,...
theorem
follows
_
If(x)
- S2m(f;x) I
and from a u x i l i a r y
I
2m f 5
[f(x)
- f(y) ] dy
m
inquality
b fly a
c) The d i s c r e t e
from e q u a l i t y
-xJ ~ d x _ <
orthogonal
I e+1
(b - a) ~ +I
systems
corresponding
to system
{hadn(X)} are the rows of matrix H2n k. D~screte systems c o n s t r u c t e d are c o m p l e t e
in R N spaces,
where
N = 2m
, 2m are orders
of H a d a m a r d
matrices. Before mation
is m o s t l y
Note lowing
investigation of use
over
used
let us note
operator
in image c o d i n g
algorithm
satisfies
the c o r r e s p o n d i n g
that
Hadamard
must
satisfy
transfor-
coding.
2-dimensionality,
and of r a p i d r e a l i z a b l e rix H a d a m a r d
item d)
in image
that the o p e r a t o r
conditions:
vanta g e s
of
existence
of reverse
method
conditions ba s e d
fol-
operator
for this pair of operators.
these
the
The mat-
and has c e r t a i n
on rapid Fourier
ad-
transfor-
149
mation. In particular, a) H a d a m a r d
transformation
of real numbers, to Fourier
w h i c h allows
transformation
b) an image code shes by s t a b i l i t y
demands
to increase
operating
received
to channel
only a d d i t i o n the rate
with c o m p l e x
by H a d a m a r d errors
and s u b s t r a c t i o n
ten times
in r e g a r d
numbers;
transformation
and by p o s s i b i l i t y
distingui-
of d e s c r e a s i n g
of bandwidth. It is n a t u r a l for every
to investigate
discrete
of t r a n s f o r m a t i o n
Hadamard
on of H a d a m a r d lation rapid
H(N)
its c o n v e n i e n t realizable
algorithm
necker
on such theorems matrices
Hadamard
for
4) but also to formuallows
to c o n s t r u c t
a
(7.10).
transformations [265]; these
factorization
were p r o v e d
a fast a l g o r i t h m
not only to c o n s t r u c t i -
which
for t r a n s f o r m a t i o n
type m a t r i c e s
to c o n s t r u c t
operations.
construction
sed on G o o d t h e o r e m [ 2 6 5 ] a b o u t Later
(7.10)
is r e l a t e d
for o r t h o g o n a l
for K r o n e c k e r
question:
of any order N = 0 ( m o d
enough
The r a p i d a l g o r i t h m s ted at first
about N l O g k N
of this q u e s t i o n
matrix
following
transformation
providing
The d i f f i c u l t y
the
were
algorithms
of K r o n e c k e r
so c a l l e d
construcare ba-
matrices.
storey-by-storey
Kro-
[176 ].
matrices
(see § 4) c o n s t i t u t e
one more class
of b a s e s
with rapid algorithms. We give Hadamard
first an a l g o r i t h m
matrix
from t h e o r e m
of order
of r a p i d H a d a m a r d
N=mn=mkn
constructed
transformation
by r e c u r r e n t
for
formula
4.5 namely:
H m n = x x Hm n-I
+ Y x H'mn_1 (7.18)
H'mn=
where
Hm n
(see lemma
-X x H'm
is an H a d a m a r d 4.3)
and
n-1
matrix
+ y x Hm n
of order
{x = {xi, j} ki,j=1
I
mn,
H'mn_1
' {Yi , j} ki,j=1 }
= -G2[Hmn-1] is a 2 - e l e m e n t a l
150
hyperframe Suppose
of o r d e r that
k.
operator
Z(mn) ] to c o r r e s p o n d
ZT Li[Z T i = ] :
L.
brings
1
with
the
the
mk [Z((i-1 k~-~ + I),
Z((i
- I) T
m
fi'
fT
bi'
ci
I)
the
+
--k--) ]
following
vectors:
f(2) .....
f(mn) ]
b Ti = L i [ b ( ] ) '
b(2) .....
b(mn) ]
T c i = L i [ c ( 1 ) , c ( 2 ) , ....
C(mn) ]
1
+ 2) .....
n
k
= L.[f(1),
1
mk
m i-J
by
Z(1) ,Z(2) ,...
vector
Z((i-
Denote
Z T =[
vector
i = 1,2,...,k
where
bi
= Hmn-1
" fi
'
ci
i=1,2,...
Here
m
n
is the
point
One
can
calculate
by
F(i)
p(i)
(7.19)
of v e c t o r
f
• f
(7.20)
formulas:
= p(i)
=
n
'
,k
transformation
F = Hm
= -G2[ bi]
+ q(i)
k Z b((j 9=I
, i = I , 2 ..... N
m - I) - - ~
+ i)
(7.21)
° X [~
k]+1,j n
151
q(i)
=
k E j:1
c((j
m ~ r.
- I)
+ i)
• Y
i-I [--~-- k]
+I ,j
(7.22)
n
where
ons
Ix] d e n o t e s
integer
Fig.
5 shows by means
for
mn
ons.
Input
point
part
of x.
of d i r e c t e d
transformation
sequence
f(1),
graph
using
the
k
mn_ I
point
f ( 2 ) , . . . , f ( m n)
is d i v i d e d
+ ~I , f ( ( i -
i) -is ÷ 2)
m
fT~ : [ f ( ( i -
sequence
i) - ~
of o p e r a t l transformati-
by
k
sequences
m
m
....
m
i=1,2,...,k
and
then
their
transformations
b
are
calculated.
Then
in a c c o r d a n -
1
ce w i t h
formulas
(7.21)
and
(7.22)
we o b t a i n
F T = IF(1) , F(2) .....
Let us estimate
the number
If we d e n o t e
by
mation
as c a n be
then,
Dn
of o p e r a t i o n s
the n u m b e r seen
f r o m the
ons
T X
and T
x and
Y
are
design
transformations
Thus,
for
numbers
for t r a n s f o r m a t i o n for
algorithm
of o p e r a t i o n s
mn
point
(7.20).
transfor-
of t r a n s f o r m a t i o n ,
+ mn
,
for m a t r i x
transformati-
respectively.
Calculation point
y
required
F(mn) ]
of o p e r a t i o n s
D n = k D n _ I + m n _ I (T x + Ty)
where
the v e c t o r
D
n
given (7.19)
we o b t a i n
c a n be u s e d and
for c a l c u l a t i o n
so on.
the e s t i m a t i o n
of
mn_ I
152
Dn
=
D o mn
(
Note
that
timation cular, and
Tx+T ~ +
1) m n k n
in g e n e r a l
of
C(k)
it w a s
after
+
k
by
found
case
C(k) ! K
factorization for
k=6,10,
factoriz~tion
-
= C(k)mnk n < C(k)NlOgkN
X
that
and
Y
mes
Hadamard
less
direct
addition
matrices and
transformation
Note
that
the
same
find X
more
and
precise
Y
es-
. In p a r t i -
C(6)
= 5,
C(I0)
=9 r e s p e c t i v e l y
, C(6)
= 4,
C(I0)
= 7.
7//
of
5
order
subtraction with
can
matrices
'
-
Fig.
For
of
one
(7.23)
N = 2 • 104 , D operations
n
= 28N
are
that
required
is
714
than
ti-
by
N 2 operations.
algorithm
can
be
used
for
reverse
transforma-
tion.
f =
I HT N m
. F n
Rapid of
Hadamard
transformation
transformations matrix
Hm
n
in
can the
be
obtained
product
of
by n+1
decomposition matrices
with
153
small
number
of n o n - z e r o
elements
n+1 H
=
H
M,.
k=1 Then sented
the
calculation
of
transformation
vector
f
can
be
repre-
in f o r m
F = M I " M2
or
of
which
is the
us
• Mn+ I • f
same)
fl
= Mn+1
" f
f2
=
f
M n
F = fn+1
Let
...
consider
the
'
,
= MI
case
(7.24)
" fn
when
the
construction
is c o n d u c t e d
by
design
H
=
XxH
(7.25)
mn Represent
H
m
mn- I
in f o r m n Hm
= n
Using
the
representation
X11H
(7.25)
=
mn
Ak
we
mn_ I
X21Hmn_ H
n+1 H k=l
I
can
X12
write
H
mn_ I
...
XI
k
X22
Hmn_ I
. . .
X2k
Xk2
H
...
Xkk
H
H
mn_ I
mn- I
X×H mn- I
Xnl
Hm
n- I
mn- I
Hmn- I
154
X11
I
X21
I
Xkl
I
mn- I
mn- I
mn- I
X12
I
...
Xlk
I
X22
Imn_ I
"'"
X2k
Imn_ I
Xk2
I
.
mn- I
ran- I
I
" " xkk
H
mn- I
ran- I
0
...
0
. ,
0
ran- I 0
H
.
ran- I =
0
0
AIS
H
...
mn- 1
Hm
S
= AI
, where
A I and
are
S
natrices
of
order
m n.
n-1 rization
each
of
blocks
H
in m a t r i x
S
we
will
have
mn- I
Hm
: A I
A2
• S
n where
A 2 = I k x X × Iron -I
Continuing
so
on,
we
obtain
,
S =
in
IkX
-th
IkXH
step:
run-1
Hm
= n
Ar
= Ik x Ik x ... x Ik x I m
x X n-r
r
An+ l =
-
I
I k × I k x ... x I k x H O n
,
r=1,2,...,n
n+1 H r=l
Ar
,
By
facto-
155
Note
that
in
special
case 1
(x
matrices
A
Now
let
by
formula
H
us
mn_ I
By
Z
be
a
Denote
M I = X x I continuation
where
with
I
n
xH
m
the
II)
[35].
the
construction
previous
(X × I
case
+ y x Z mn- I
is we
obtain
)S, mn_ I
matrix
of
order
0
t
0
0
...
0
0
-1
0
0
0
...
0
0
0
0
0
1
...
0
0
0
0-1
0
...
0
0
0
0
0
0
...
0
I
0
0
0
0
...-1
0
, then
H
mn- I
m
n
in
= M I
form
S
.
mn
factorization
for
each
block
of
H mn- I
S
we
obtain
that
Hm
= M I
M2
•
S
,
n where
M2
=
I k x (X x I
+ Y x Z ran- 2
S
=
conducted
n-1
+ y x Z of
o
case
mn- I By
0
the
commutative
Z =
9
II o
matrices
analogy
=
~ =
Good
+ y x H' = mn- I
S
Let
,
with
consider
= X × H
I
It 1 _lII
coincide
r
(7.16).
mn
=
I k x Ik x Hmn_2
) mn_ 2
,
matrix
156
Hence,
Hm
n+1 U Mr r=1
= n
where
Mr
= I k x Ik x "'" × Ik x (x × Im
+ Y x Zm n-r
r -
)
'
n-r
I
M n + I = I k x I k × ... x I k x H O
, r=1 , 2 , . . . , n
n
we der
m
obtain and
n
After quential
that
= N
row
is
F
can
~ n+1
m,
for
r
= n+1
it
is
• m
k
° m
we
order
Hadamard taken
and
constructed order
of
of P2
~i
times
order are
operations
for
i#I
In t h e
Po
the
formula
H2
+ n.k.m n
is
and
for
general
ek Pk
order
and
m.m n
algorithm
"'"
as
non-zero
of
elements
of
f
is
in
l
or-
se-
= mn(m+kn ) =
n+1 U P. i=I l
. p~l
PoP~ l
taken
using
(7.18).
matrices
needs
i=I
=
square
calculation
for
rapid
of
for
operations
the
form
H°
one
N
= PoP~ l
matrix
n
in n÷1 n n u m b e r of
that
where
construct
N
n
calculated
Hence,
fix
clear
(7.24)
m
m
the
r
r
operations,
= m'k n
M
H
for
made
be
of
k,
n+1 Z Pi i=I
of
of
transformations
N
of
decomposition
in e a c h
notes
order
are
the
Hadamard
matrix
case
Hadamard
constructed
hyperframe{
(7.18)
an
initial
of
order
as
Then
mat-
follows.
X I , Y I} o f
is u s e d . one, ~I PoPI
the
of
order
PI'
H I matrix
hyperframe { X 2 , Y 2} ~2 P2 is c o n s t r u c t e d
157
using der
(7.18) a n d s o on. It is e a s y t o ~1 ~2 ok PoP1 P2 "'" P k can be also
method
and
that
for
these
Dn = N
rix
Taking
into
M i can
be
that
Hadamard
factorizated
by
matrix
of
or-
abovementioned
matrices
k Z eiPi ~N i=1
account
see
that
factorizated,
k X Pilogp N i=I i
in d e c o m p o s i t i o n we
can
obtain
more
(7.26)
n+l H M. each mati=I l precise value for H =
(7.26)
k
Dn = N
where
C(Pi)
are
constants
(7.27)
~ eiC(Pi) i i=I
depending
on
construction
{Xi,Y i} of order Pi" Finally, EXAMPLE
let us I. L e t 1 -1
-1
give
two
matrices
examples. X,Y
be
of
form
-1
-1
-1
1
0
0
0
0
1 -1
-1
-1
0
1
0
0
0
1 -1
-1
0
0
1
0
0
1 -1
0
0
0
1
0
0
0
0
0
I
I
I
I
I
I
I
I
I
I 1
-1
-1
-1
-1
-I
-I
-1
-1
-I
I
0
0
0
0 -I
0
I
0
0
0
I -I
0
0
I
0
0
I
I -I
0
0
0
I
0
I
1
I -I
0
0
0
0
I
I
I
1
X =
1
I -I
of h y p e r f r a m e
158
y
=
0
0
0
0
0
0
1 -1
0
0
0
0
0
1
0
1 -1
0
0
0
0
0 -1
1
0
1 -1
0
0
0
0
0
1
0
1
0
0
0
0
0
1
0
1
1 -1
0
0
0
0
0
1
1
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
that
X
and
0 -1 -1
0 -1
1 -1 1 -1
It is e a s y frame
of o r d e r
common
number
ced
0 1 -1 1
0 1 -t
to v e r i f y
-1 0
-1
1 -1
10. As a r e s u l t of o p e r a t i o n s
f r o m 80 to 60;
of t r a n s f o r m a t i o n s
!
-1
-1
C(19)
has
-1
-1
1 -1
Y
f o r m the
of f a c t o r i z a t i o n
in b l o c k s reduced
factorizated
X
,
F i g .6
X
and
of m a t r i c e s Y
respectively and
Y
2-elemental
(see fig.5) from
is s h o w n
hyper-
X has
and
redu-
9 to 7. T h e
in fig. 6
Y,
graph
159
EXAMPLE
2. Let us give
transformation tions
of order
for a m a t r i x
the c o m p l e t e
12 d e m a n d i n g
algorithm
60 a d d i t i o n
of rapid
Hadamard
and s u b t r a c t i o n
of type
H = X 1 × H 1 + X2 x H2
w here
-t
1
0
0
0
0
0
1 -1
0
1
0
0
0
0
0
0
1
0
0
0 -1
1
0
1
1
1
1
0
1
1
1 -1 -1
x£
HI
-1
-1
1
,
1
0
0 -1
0
1
0
1 -1
0
0
1
1
and
H2
Fig.7
1
X 2
shows
transformation
matrices
the graph of c o m p l e t e of order
H1:11
1 q 1 -1
-1
-1
0 -1 -1
0
0
0
0
1
0
0
0
0
0
0
0
of order
2.
=
1 -1
are any H a d a m a r d
0 -1
algorithm
12 for
I[
,
H2
=
[I 1 -1 -1 -1
&
~o &
,H
FL~/.?.
II
of rapid H a d a m a r d
opera-
160
Now consider statement
4.11.
the f a c t o r i z a t i o n Represent
storey Kronecker
matrix
Hk =
Ak
matrix
of H a d a m a r d Hk from
matrices
(4.60)
obtained
in
in f o r m of s t o r e y - b y -
[176].
=
J PI x Ak-1 + P2 × Bk-1 R I x Ak_ 2 + R 2 x Bh_ I
(7.28)
where
s I =lIPIII
,
s 2 =liP211
RI
Let us d e n o t e by Mi,
I i-1 =
•
M
Ini-t
~ IIR1,R2tl
Mk+1
f o r m of p r o d u c t
The m a t r i c e s of
k+1
"k = M I PROOF.
Hk=
=
Bo
the m a t r i c e s
~ IIPI'P2 II
n
7.9.
H o = lIAoll
i=I,2,...,k+I
=
THEOREM
,
R2
Matrices
Hk
xI
mn
, i=1,2,...,k
k-1
I k x AO n I k x BO n Hk
from
(7.28)
sparse m a t r i c e s
"M2
"""
~
in f o r m
+ (P2 x I k_1) (In x Bk_1) I + (R 2 x imn k _i ) (I n x Bk_1) mn
irPix IP2xlkl IInXkl Imnk-
RI
k-1
x I
mn
mn
R2 x I
k-1 mn
in
[176]:
" ~+1
c a n be r e p r e s e n t
(PI x I k_1) (I n x Ak_1) (R I x Imnk_1) (I n x Ak_1) mn
are r e p r e s e n t a b l e
I n x Bk_ I
(7.29)
161
Further
manipulations
with
Ak_ 1
and Bk_ 1 allow to obtain
the
representation
P1 H k = (If RI
P2 I n x II PI,P211 R211 x Imnk_1 ) (If In × II R1,R211 x Imnk_2)"*"
I k-1 (If n
xll PI'P211
k-1
xl] R I,R21j
n
fink × A IIx Im) ( 1 °II
demands
Rapid Hadamard tant properties:
transformations
properties,
THEOREM
satisfy
(n) Gi,j = •
7.10.
G i,j (n)
PoP~
(I)
(Am, j
(7.29)
mentioned
have the following (hybrid)
for new types
orthogonal
matrices
H
n
bases
the ortimal
of covariant
is based on the following
impor-
matrices
G.
theorem.
are eigenvectors
of matri-
(n) Pl -I = {Gi,j }i,j=0
G(n)
blocks
by decomposition
b) they allow to obtain
2 - 4
Let the rows of Hadamard ces
of rapid Hadamard
operations.
of problems
The latter property
Mk'Mk+ I
specified
a) they allow to construct
with predetermined solutions
as for first algorithm
that the algorithm
=(Nlog N)
M2"...'
n
It is easy to calculate, transformation,
) = MI
I k XBo
(Matevosyan,
of matrices
Suppose
that elements,
G (n) = {G i,j (n) } PI-1 i,j=0
' n ~ I
the
of order
the relations
P1~ I (I) I(n) (Xm i m=0
• I(n)
1984).
(2)
+ Xm, j
Then the eigenfunctions
+ X (2) m,i
zT(n))
G(n-1) m
Z(n))
(7.30)
of these matrices
are the functions
of Hada-
mard base H . n PROOF.
From the orthogonality
of rows of Hadamard
matrices
H
n
162 and from the p r o p e r t i e s G(n)
PI~ I [m]=0
=
of the h y p e r f r a m e
(X(1) [i],[m]"
we can o b t a i n
2) . zT) . H T I + Xli],[m ] n-1 PI-I
x
(XlI)
. (2)
m],[j]D[m]Hn_1
+ XI21
,
PI -
[m]
n-1
,(t) (~[i],[m]"
(I)
x
Dm
(I) Hn-1 × ( X [ m ] , [ j ]
I + X I 2)i],[m]
I + A[m],[j ]
let us c o n s i d e r signal
te signal
" I + X (2)
+
•
Z)
=
obtains
a
[m],[j]
Z T) x G(n-1) [m]
x
Z
X(n) with
Z)
the Wyner
= Y(n)
filter
+ Z(n)
zero average
input of w h i c h
representing
and noise
y
ponents.
We assume
that the a u t o c o r r e l a t i o n
matrices
R xx, RXZ;
R zz
transformation transformation
A
are k n o w n
is p e r f o r m e d
is p e r f o r m e d
~i" The p r o b l e m
base
(Wyner
filter)
ding
the best r o o t - m e a n - s q u a r e
in the choice
weights
are o b t a i n e d
i,i [AR xx A -I ] i,i of e s t i m a t i o n
[ARXZ A-I 2 = Tr(RZZ ) -
E i
and then a r e v e r s e multiplied
filtration
of l e g i t i m a t e
1
error
com-
an o r t h o g o n a l
with
of the w e i g h t s
[AR xz A -I ]
for r o o t - m e a n - s q u a r e
filter
components
linear
estimation
sum of l e g i t i m a -
and c r o s s - c o r r e l a t i o n
input vector
of o p t i m a l
consists
that the o p t i m a l
for
the
with n o n - c o r r e l a t e d
[194].In W y n e r
for spectral
me w e i g h t s
is k n o w n
1) (X i],[m]I
E
[m]=0
. (2)
x (Xrml,ri1"t J L J
random
x
Z) =
I
E [m]=0
Now
+ ~[m],[j]D[m]Hn_1
ZT) x HT
that
2 ]i,i
[AR xx A - 1 ] i , i
limited
~i p r o v i -
signal
by formula
by so-
Z. It
163 The result
is g e n e r a l i z a t i o n
Optimal
Wyner
by a u x i l i a r y
filter
culations ons.
over all bases.
filter.
Howevwr,
realization
Generally,
suboptimal
filters
against
Numerical
a background
mation
tends
to the error
2) the q u a l i t y
error
rier and K a r u n e n - L o e v e
of the
filter
filters
have
differs
although
use
filter
is
for the cal-
arithmetic Welsh,
operati-
Hear t r a n s f o r of M a r k o v
sig-
shown that
filter
filter
is o b t a i n e d
of such
for e x t r a c t i o n
noise
of optimal
of Welsh
2N2+N
with Fourier,
of white
[24].
on the base
An example
of
experiments
I) the r o o t - m e a n - s q u a r e
equation
it is of little
it demands
are used.
Wyner
limitations
since
mations nals
without
optimization
the K a r u n e n - L o e v e
of classic
with
Fourier
asymptotic slightly
at
transfor-
N ~ ~.
from that of Fou-
it a p p r o a c h e s
to other
value
[24]. Fourier
and W e l s h
transformation realization. rences
transformations
is more p r e f e r a b l e
The H a d a m a r d
relative
ference
relative
vectors
with
to Welsh
dimension
because
discussed
to Fourier
have
of rate
algorithms
and Welsh
and s i m p l i c i t y
in this work has a n a l o g o u s
transformation transformation
not equal
rapid
2 k and
and b e s i d e s that
it has
of
prefe-
such pre-
it can be r e a l i z e d
it is optimal
over
for new class
of processes. Finally,
let us give
lized H a d a m a r d
transformations
zed H a d a m a r d
matrices
se two times
the number
classic
the a l g o r i t h m s
constructed
in chapter
Fourier
3-dimensional
on 2- and 3 - d i m e n s i o n a l
of o p e r a t i o n s
2- and 3 - d i m e n s i o n a l
I. The a l g o r i t h m
based
of 2- and
2. They
required
genera-
generali-
a l l o w to decrea-
for c a l c u l a t i o n s
of
transformations.
of t w o - d i m e n s i o n a l
generalized
Hadamard
transfor-
mation. Let
[X] 2
be a cubic Rewrite array
be a t w o - d i m e n s i o n a l
generalized
Hadamard
the t w o - d i m e n s i o n a l
[X] 2
in the
following
input array
matrix
generalized form
of order N and
[W] 3 =ll Wijkl] Hadamard
[W] 3
=II y~(i+j)i[
transformation
of
164
(7.31)
[A] 2 = 1'1 ([W] 3 -[X] 2)
or
(what is the same) N-I =If kZ0Wijk=
[A] 2 =il Ai,jll
N-I k(i+j) = II k=0ZYN
Xkjll
(7,32)
Xk,j i[
So, we obtain N transformations N-I
I! Ai,o{{
where
F*
= II z yk,i Xk k=0
is adjoint-transpose
li Ai,ji i = (F*U jT)
,o
II = F*X
of Fourier
" Xj,
(7.33)
o
matrix
F.
(7.34)
j=1,2 ..... N-I
where
[X] 2 = (Xo, Xl,..., The reverse
transformation
1
Transformation
([W]31
(7.33)
transformation,
based on the matrices F*
(7.31)
is
1,1
Ix] 2 = ~
Fourier
for
XN_I) •
,[A] 2 ) , [W]71
is the ordinary
and N-1 discrete obtained
: [W3 ]
(i,k)
one-dimensional transformations
from Fourier
discrete (7.34)
are
matrix by permutations
" U jT. The number
of operations
tion of fast transformation little
as for realization
formation
(multiplications) from
(7.33)
required
is N21ogN
of known two-dimensional
that
for realizais twice
fast Fourier
as trans-
~65].
2. The algorithm
of three-dimensional
generalized
Hadamard
trans-
165
formation. Using above-mentioned Hadamard
transformation
three-dimensional Let
[X] 3
algorithm
of t w o - d i m e n s i o n a l
we w i l l c o n s t r u c t
Hadamard
be a c u b i c
an e f f e c t i v e
Hadamard
de the i n p u t a r r a y
Hadamard
N = 3 T 3. D e f i n e
matrix
[X] 3
algorithm
for
transformation. m a t r i x of o r d e r
[B] 3 = {yN(i+k) (i+J) }, i , j , k = 0 , 1 , . . . , N - 1 bic g e n e r a l i z e d
generalized
into
M = 3T 2
and
, be a c o m p l e t e l y
proper
cu-
[H(N,N) ] 3 of o r d e r N = 3 T 3. Let us d i v i -
3 square m a t r i c e s
for t h e m 3 t w o - d i m e n s i o n a l
Z3, i = 0 , I , 2
generalized
of o r d e r
Hadamard
transfor-
mations
The r e v e r s e
_
Zo
Yo = I'1([B]3Zo)
(7.35)
li,j,k YI = 1,1 ([B]~k,i,J)Z1)
(7.36)
i,j,k (j,k i ) Y2 = I'I([B]3 ' Z 2)
(7.37)
transformations
I 1,1 N ([B] 3 Yo )
1 1,1 '-I ZI = ~ ([B] 3
for t h e m are r e s p e c t i v e l y ,
(i,k)
, where
YI ) , w h e r e
1 1,1 . . . . . Z2 = N ([B] 3 -I y2 ) , w h e r e
By a n a l o g y divided
y
into
w i t h the p r e v i o u s N
following
(o) = F*Z o
Yo
o (I) = F * R Z
[B]~ I
=[B]3
'-I [B] 3
=[B]3
I
(I)
'
y
3
=[~]
[B] 3
algorithm
(i,j)
3
e a c h of
(7.35)
-
(7.37)
transformations
(o) , y.,O,(% = U ~,F , u~J~T z ,o 1% 3 3
o
(j,k)
, j=1,2 ..... N-I
(I) = u J T ( F . R ) u J T z . ( 1 ) 3
'
j=1,2
'"
,
..,N-I
'
is
166
y
(2)
= RF*Z
o
where
i=0 "'"
•
R
is the
1'''"
,y
,N-I
(2)
3N21ogN
matrix
algorithm
+ 2N(N-I)
operations
of fast
transformation
[(3T3-1)/2]
of o r d e r
N
with
required
M
elements
• Yj
,
i2 YN
(Yo (j) ' YI (j)
=
'
give
and
some
of c o m b i n a t o r i a l
defining
arrangement
on of
stringent
ments,
their
pairs•
interrelations t - designs, F-square
Youden
adopted
noiseless
designs
Hadamard
other
M=3T 3 demands
+ 27T61og3
+ 6T 6 - 6T 6 +
3-dimensional
fast
Fourier
theory
of H a d a m a r d
i.e.
in some
hand.
subsets
Note
that
theory
preservati-
We c o n s i d e r
designs these
of e x p e r i m e n t s ,
automaton
with
on the one hand,
factor
of e l e here
the
and between
and orthogonal designs
graph
(Markova
in
structures
of the o c c u r e n c e s
combinations.
matrices
matrices
mathematical
the n u m b e r
partial
for p l a n n i n g
coding,
generali-
of o r d e r
applications
and other
designs,
on the
of
of e l e m e n t s
trios
3-dimensional
3M31ogM = 2"82T61ogT + 81T61og3
design
concerning
between
designs
widely
theory,
rules
of
.
construction the
= 81T61ogT
for c a l c u l a t i o n
matrices
In § 8 we w i l l
IX] 3
i n s t e a d of
of o r d e r
§ 8. H a d a m a r d
calculation
of a r r a y
+ [ (N-I)/2]
operations
transformation
1979,
(2) , j = I , 2 , . . . , N - I
3
(Zo(J) •Z I (j) "'*" ,Z N-I(J))
=
the
zed H a d a m a r d
turn
= UJ(RF,)UjZ
]
diagonal
Z3
'
y
#
(J)) 9=0•1• 2 N-1 '
Hence,
+
(2) o
are
theory,
E.V.,
Ezova
in group L.I.,
1981).
8.1.
H adamard
matrices
and t-designs.
L e t us g i v e
the
following
definition. DEFINITION
8.1
[61]
. Let
X be
a set of v e l e m e n t s .
Define
a t-de-
167
sign
with
subsets
parameters (blocks)
- every
NOTE
t
system
symmetric
k=r
the
family
following
of
b different
conditions:
different
blocks;
in p r e c i s e l y
I blocks.
is
by
order
incomplete
R
r
appears
k-elemental
of
a
subsets
S(t,k,v) 2 at
of
at
i =2,
block-design
some
i =
I,
set
at
i = 0;
v=b,k=2;
v=b=n2+n+1,
SBIB
at
k=r=n+1,n~2.
t=2,
v=b
(and
hence,
too). Note
question lis
X
(denoted
plane
be
in p r e c i s e l y
t-design of
to
satisfying
from
collection
- projective
X
occurs
[40].
- Stainer
-
set
subset
8.1.
mere
-
of
element
each
-
t-(v,b,k,r,l)
that is
and
most
the
1972)
ting
applications
John
(1974), let
matrices
and
THEOREM necessary
with
of
give
of
the
besides
and main
unsolved
t-designs
different
t-designs
J.McWilliams us
and
construction
Street,
Now
important
(Hall,
parameters
one
can
Sloane
in
find
general
1970, and
case
J.Wallis,
that
in p a p e r s
some
of
Wal-
interes-
Endate
and
(1979).
results
about
relations
between
matrix
of
order
design
with
Hadamard
t-designs. 8.1.
and
For
existence
sufficient
of
Hadamard
exastence
of
SBIB
a)
v
= b
= 4t
-
I
,
r = k
= 2t
-
b)
v
= b
= 4t
-
I
,
r = k
= 2t
,
I
,
i = t -
4t
it
is
parameters I
,
or
COROLLARY then
there
8.1.
exist
Let BIB
there
exists
designs
with
an
i = t
Hadamard
.
matrix
of
parameters
a)
v
= 2t-I,
b
= 4t-2,
r
= 2t-2,
k
= t-l,1
b)
v
= 2t,
b
= 4t-2,
r = 2t-I,
k
: t,
c)
v
= 2t-I,
b
= 4t-2,
r = 2t,
k
= t,
= t-2
I=
t-2
I = t.
;
;
order
4t;
168
COROLLARY exists
8.2.
the Hadamard
Information ce
sets o n e
(1967),
find
non-isomorphic rillary
8.1.
Hadamard
So,
view
the
Singhi
designs
also
that Bhat
Hadamard
matrices
Turyn
differen-
(1965) , H a l l
(1971),
matrix
Wallis
and
one can construct from theorem
that each
of
to construct
Singhi
also
there
(1976).
shown
20 a l l o w s
are
Then
of different
(1963),
Szekeres
Dillon
has
and
papers
Note
has constructed
(1974)
two non-isomorphic that one can
BIB designs.
From
of
Shrikhande
in p a p e r s
this point
(1971a),
of n e w o r d e r s
and co-
3 non-isomorphic
has proved
interest:
8.1
several
using
of
(1970),
(1971b),(1971c) theory
of
finite
plane.
THEOREM order
(1967),
such non-isomorphic
(1975).
projective
and relations
with parameters
(1972)
set.
4t.
an Hadamard
of order
following
be a d i f f e r e n c e
of R y s e r
(1975),
designs
Bhat
(19,19,9,9,4) only
Storer
Having
BIB
matrices
construct
in p a p e r s
Spence
8.3.
of o r d e r
applications
(1966),
(1972),
COROLLARY
(4t-1,2t-1,t-1)
matrix
about
can
Jonson
Whiteman
BIB
Let
8.2.
[293].
For existence
4n 2 it is n e c e s s a r y
and
of r e g u l a r
sufficient
Hadamard
existence
matrix
of
of SBIB designs
with parameters a)
v = b = 4n 2, r = k = 2 n 2 - n ,
~ = n2-n
b)
v = b = 4n 2, r = k = 2 n 2 + n ,
~ = n2+n
or
COROLLARY Then
there
8.4.
exists
Let
there
an SBIB
exists
design
an Hadamard
with
matrix
of o r d e r
4n 2 .
parameters
a) v = b = 16n 2, r = k = 8 n 2 - 2 n ,
~ = 4n2-2n
b) v = b = 16n 2, r = k = 8 n 2 + 2 n ,
~ = 4n2+2n
or
THEOREM there
exists
8.3.
[40].
a 3-design
Let
H be an H a d a m a r d
with parameters
matrix
of order
4t.
Then
169
a)
v = 4t,
b = 8t-2,
b)
v = b = 4t,
k = 2t,
r = 4t-1,
I = t-1
or
J.Wallis theorem
(1970)
8.3
quasi-symmetric
in p a p e r s
of S t a n t o n
DEFINITION can extend
has proved
I = t-1.
hhat
is a q u a s i - s y m m e t r i c
tion between find
r = k = 2t,
8.2.
the
A
matrix
wit parameters
[61].
Information
and Hadamard
(1968,
t-design
incident
design
design
e t al
3-design
matrix
a)
from
about
one
rela-
can also
1969).
is c a l l e d
an Hadamard
of the d e s i g n
design
to d e f i n i t i o n
if o n e
of H a d a m a r d
matrix. Note mard
that
3-design
with
parameters
b)
of
theorem
8.3
is the H a d a -
design. DEFINITION
8.3.
A
(t-1)-design
D
is c a l l e d
a derivative
of t-
P design D
D
in p o i n t
and blocks
of
DEFINITION tive
of
sign
(different
no points
[ 61]. A t - d e s i g n Dp,
of
if the
D
D) p o i n t s
of
p) .
is c a l l e d
design
from
Dp
an extended
is i s o m o r p h i c
deriva-
to t - d e -
D.
dable,
8.4.
[61].
then one of the
I. D
If a s y m m e t r i c following
is an H a d a m a r d
2. v =
k = 11,1
4. v = 495,
k = 39,
8.2.
2-design THEOREM
8.5.
2-design, I. D 2. V =
If
D
is e x t e n -
is t r u e
=I i= 3.
1
the u n i q u e 3-(v,k,1)
t h e n o n e of t h e
is a n H a d a m a r d
conditions
design
k =12+31+I
From condition allows
2-(v,k,1)
2-design.
(I+2) ( 1 2 + 4 1 + 2 ) ,
3. v = 111,
NOTE
ric
8.4.
(containing
(t-1)-design
THEOREM
mard
p
p if it c o n s i s t s
of t h e o r e m
k =
follows
that
an H a d a -
extension. design
following
D
is an e x t e n s i o n conditions
3-design.
(I+I) ( 1 2 + 5 1 + 5 ) ,
8.4
(i+I) (i+2)
is t r u e
of
symmet-
170
3. v = 112,
k = 11,1
= I
4. v = 436,
k = I0,I
=3
Finally the
let us n o t e
theorems
8.3,8.4
(1980)
and p r o o f
Lint
of J . W a l l i s , W a l l i s
8.2.
DEFINITION be c a l l e d and
in
designs
square
of
item
in w h i c h
8.1
have
ordered
then
there
STATEMENT
8.2.
If t h e r e
there
struction. gements
Kiefer of H a d a m a r d
(1975a), The
For
can
find
(rectangles)
consisting
of e l e m e n t s
in p a p e r
.
of e l e m e n t s
v
on the
of vxk
with
pair
appears
X
will
in e a c h
row
set of e l e m e n t s
Youden
parameters
are
design
X.
is e q u i v a l e n t
(v,k) . M o r e o v e r ,
algorithms
the
subsets
of d i f f e r e n t
taking
into
(2t2-1)x(4t) (1975b)
if
has
still
transforming
of rows
symbols
of L a t i n
appears
consideration
among
matrices v=4t, other
Hadamard
a regular and
the
in the
results
f2~2,
constructions
matrix
of o r d e r
4t.
of o r d e r
(2t2+t)x(4t29
Youden
designs.
the c o n c e p t
are
arran-
are c o n s i d e r e d
shown
that
the e x i s t e n c e
the e x i s t e n c e
b1=2t(4t-1) obtained
con-
block-designs.
has
implies
of Y o u d e n
blXb 2 rectangular
the c o l u m n s
are b a l a n c e d
of o r d e r
(2t)x(4t-1)
matrix
and
results,
or
Hadamard
designs
the r o w s
other
f1>0,
(2t-1)×(4t-1)
generalized
Youden
(when
a
a regular
exists
arrangements
(1975b),
the
exists
exists
generalized
these
of g e n e r a l i z e d signs.
exist
of e l e m e n t s
as blocks)
one
J.van
we o b t a i n
design
Kiefer
designs
constructed
Hence,
If t h e r e
then
vxk
designs
8.1.
Youden
8.2
and
of
designs.
Youden
each
and proof
of P . C a m e r o n
and
of B I B d e s i g n
design
Smith
of c o l u m n s .
STATEMENT
4t 2,
SBIB
and
that
and y o u d e n
existence
into Y o u d e n
also
number
of
the
8.1
- 8.3
(1972).
if e a c h
that
Note
same
design,
8.1
in p a p e r
theorems
A rectangle
It is k n o w n
Hartley
find
Street
are b l o c k s
1948
SBIB
of the
the c o l u m n s
to the e x i s t e n c e
the d e f i n i t i o n s
can
matrices
8.5.
Youden
one
and
Hadamard
that
(2fi-I),
of a s e r i e s
i=I,1,
from Hadamard
Youden
matrices
de-
171
and g e n e r a l i z e d Finally, Youden E.V.
(1975),
Markova
E.V.
orthoginal
F-square
information
about
p l a n s one can
matrix
elements
the reader
L.N.
of order
to Kiefer
in p a p e r s
berween
of M a r k o v a
(1981).
n
F-square and
desi@ns.
~ =
Let
(x1,~2,...,Im)
Let also v = { 1 1 , 1 2 , . . . , i m} be an o r d e r e d
from
design
A. The m a t r i x
or an f r e q u e n c y
A
will be c a l l e d
square
in each row and each c o l u m n
of m a t r i x
A
an
and will be d e n o -
ted by F ( n , l l , 1 2 , . . . , l m ) , if for e a c h n, n = 1 , 2 , . . . , m ars
(1975b).
the r e l a t i o n s
find
and o r t h o ~ o n a l
be a vector.
set of d i f f e r e n t
we refer
and Ezova
matrices
be a square
~I+~2 + ' ' ' + I m = n
that
and d i f f e r e n t
H adamard
A ={ai,j}
designs
let us note
designs
8.3.
Youden
element
precisely
Ik'
c k appelk ~ I
times. Note
that F(n,1,1,...,1)
Information designs
and Ezova
THEOREM re ex i s t
give
8.6.
(4t-I) 2
Finally,
only a t h e o r e m
assembled
Let
and the a p p l i c a t i o n s one can (1975),
find
n
.
of o r t h o g o n a l
in p a p e r s
Kirton
of
and Seberry
about r e l a t i o n
be an H a d a m a r d
between
that an e x t e n s i v e
Hedayat (1978),Mar-
Hadamard
of order
square
matri-
4t.
Then
the-
designs.
survey on r e l a t i o n s
and o r t h o g o n a l
between
designs
Ha-
has been
(1978).
of H a d a m a r d
we o u t l i n e
of e x t r e m a l
independent
factor
and Wallis
applications
matrix
F(4t,2t,2t)
and p a r t i a l l y
In this p a r a g r a p h in p r o b l e m
H
orthogonal
by H e d a y a t
§ 9. Other
of order
F-designs.
we note
damard matrices
et al
square
(1981).
ces and o r t h o g o n a l
pairwise
theory
(1970),Hedayat
We will
ces
the
and their g e n e r a l i z a t i o n s
and Seiden kova
about
is a Latin
matrices
some a p p l i c a t i o n s
geometrical
r a n d o m variables,
constants, Barker
of H a d a m a r d in m a x i m a l
sequences,
matri-
sets of
strongly
re-
172
gular graphs,
in m a x i m u m
9.1. H a d a m a r d
determinant
matrices
The t o p i c of e x t r e m a l
problems
and a extremal geometrical
a n d in w e i g h t e d
~eometrica!
constants
numerical
characteristics
tial description
of v e c t o r
s y s t e m s w h i c h are e x t r e m a l
some p r o p e r t i e s
[ 230,
Let
X
constant.
consists
t i o n of e x t r e m a l
of v e c t o r
plans.
of c a l c u l a -
systems
a n d spa-
in r e g a r d to
252].
be a l i n e a r n o r m a l i z e d
space over R I, i~
, 1
be s p a c e s P
of n u m e r i c a l
sequences
II x
a
= {Xl,X2,...,}
(vectors) if
x =(Xl,X2,...)
II : suplxil i
,
be n o n - o r d e r e d
x i of the
I[ x I I = ( XlXr I p ) I / p i
collections
space X. W r i t t i n g
a m = { X l , X 2 , . . . , x m}
, then
Va k
consists
3alCan
, means that , where
Jl all
= II (q) II •
o = a(1,k,x)
such that
II xill ~ I, i:I,2 ..... n } c X
we h a v e to c a l c u l a t e
~(l,n,x)
~n c X, 9.1.[230].
inf X
THEOREM
,
the m a x i m a l
of the p o i n t s
: II a III _> 6
In o t h e r w o r d s ,
THEOREM
q n c Om, n < m
(q) = gx~g a n d
in f i n d i n g
: { { x 1 ' x 2 ..... X n}
(systems)
qn = {Xil,Xi2,...,Xin}
I ~ i I ~ i 2 ~ i 3 ~ ... ~ i n ~ m. F i n a l l y , The p r o b l e m
w i t h the n o r m s
9.2
Let
the c o n s t a n t
= inf max]I OlI 1
01 c Ok,
II xi11 _> I
k > i, t h e n
~(l,n,X)
= 6(l,k,l
[252] • The
following
) = 1/(21-I)
statements
are e q u i v a l e n t
173
I. 6(2,4n-I,11
4n-I
) = (4n-2)/(4n-1)
2. ~(2,4n,i I4n-1)
= (4n-2)/(4n-1)
3. There exists an H a d a m a r d m a t r i x of order 4n.
9.2. H a d a m a r d m a t r i c e s and Barker x
n
Suppose Xl,X2,...,
is any sequence of complex numbers. D E F I N I T I O N 9.1.
[134]. A sequence C I , C 2 , . . . , C n _ I
=
C3 where X cT
n ~ J x l x C T (i + j ) i=I
is the c o m p l e x c o n j u g a t e of
ce of length
n
provided
Note that the sequence cal
se~uencgs.
mod
X
C. 6 {0,-1,+I} 3
,
n
is called a Barker
sequen-
, j=1,2,...,n-1.
{Cj }n-lj=1 ' C 3' £ { - 1 , + 1 }
is
used
in
numeri-
c o m m u n i c a t i o n theory. Turyn and Storer
length
S > 13
(1961) have proved that the Barker sequence of
can exist if and only if there exists a c i r c u l a n t
(hence, regular)
H a d a m a r d matrix of order
ce of length
can also exist only
S
n. Thus,
the Barker sequen-
for s=k 2.
9.3. H a d a m a r d m a t r i c e s and stron~!y regular graphs. A graph is c a l l e d regular g r a p h of power
d
G
if the powers of all verties are
d. In 1963 Bose i n t r o d u c e d a n o t a t i o n of strongly regular graph G = (n,d,A,A)
of power
d that is a graph every two n o n - a d j a c e n t ver-
tices of which are s i m u l t a n e o u s l y a d j a c e n t to ~ v e r t i c e s and every two a d j a c e n t v e r t i c e s are s i m u l t a n e o u s l y a d j a c e n t to A vertices. Note that in G(n,d,A,
), n is the number of vertices,
number of triangles,
A
d is the power,
A
is the
is the number of plugs [329].Information about
strongly regular graphs and their r e l a t i o n s to the c o m b i n a t o r i a l f i g u r a t i o n s one can find in papers of Bose
(1959),
(1963), Seidel
con-
174
(1967-1969), A l i e v et al
(1969), W a l l i s
del
(1972), Delsarte
(1970), Wallis et al
K o z y r e v V.P.
(1969,
1971), Goethals,
(1972),
Zinovjev V.A.
Sei~ and
(1975).
Here we will give only 4 G o e t h a l s - S e i d e l
theorems
(1970) about
the c o n n e c t i o n b e t w e e n H a d a m a r d m a t r i c e s and strongly regular graphs. Note that for strongly regular graphs one can find three e i g e n v a l u e s from the r e l a t i o n s
[329]
I O = d , 11, 2 = ~I(A-A+_ V(A-A) 2
4A+4d
T H E O R E M 9.3. A symmetric H a d a m a r d m a t r i x the c o n s t a n t diagonal of order
s
2
exists
H = A~I, A T = A with
if and only if there exists
a regular graph w i t h e i g e n v a l u e s
11 = 2s Z I,
12 = -2s ~ I
Note that first part of the t h e o r e m is introduced by Menon
[329] .
T H E O R E M 9.4. A regular symmetric H a d a m a r d m a t r i x with the constant d i a g o n a l of order Ls(2 s ) [ 3 2 9 ] or NLs(2 T H E O R E M 9.5.
4s 2
exists
s)[329].
If there exist a BIB design with p a r a m e t e r s v,k,r,
I=I and an H a d a m a r d m a t r i x of order r e g u l a r graph with v+k-1
v(m+1)
then there exist a strongly
v e r t i c e s and with the e i g e n v a l u e s
11 =
If there exist a finite p r o j e c t i v e plane PG(2,m-1)
and an H a d a m a r d m a t r i x of order regular graph with =
m
and 12 = -m.
T H E O R E M 9.6.
i°
if and only if there exist graphs
0,
11
=
m
2
-m+1,
m(m2-m+1) 12
=
m+1
then there exists a strongly
v e r t i c e s and with the e i g e n v a l u e s
-m.
9.4. H a d a m a r d m a t r i c e s and m a x i m u m d e t e r m i n a n t p rpblems. A = {ai, j }n i,j=1
is a real m a t r i x and let
SuppQse
175
= maxldet
A
, for
a
f(n)
= maxldet
A
, for
a
g(n)
= maxldet
A
, for
a
k(n)
= maxldet
A
, for
0 < a.
h(n)
= maxldet
A
,
for
6 {0,1}
1,3
6 {0,-I,+I}
.
1,3
-
l(n)
6 {-I,+I}
1,3
that
is a l l m e n t i o n e d are < -
equivalent
h(n)
problems [171].
= g(n) namely,
In
-
-I < a. -
It is k n o w n
< I
1,3
= k(n)
< I.
lw]
= l(n)
= 2n-lf(n-1),
calculations
1893 H a d a m a r d
-
that
h(n),g(n),...,l(n)
proved
that
h(n)
<
2-n (n+1) (n+1)/2 ~ and
there
exists
h(n) < n n/2 Note
only
an H a d a m a r d
= I,
Information (1931),
(1963),
(1973)
I, 2,
about Bellman
Yang
Hadamard
weighted
plans
(weight,
length,
t i o n of c h e m i c a l
(1943),
accepted
frequency elements
one)
32,
remain
10,
11,
144,
12,
320,
1458,
one can
Williamson
(1944),
Ehlich
(1970),
Cummings
- statistical
so on)
several
determine
4),
unknown.
13
problem
aspect
plans.
voltage,
there
find
9477
in
(1964), Payne
of the p r o b l e m ) .
It is k n o w n where
The
together
individual
that
the
a measure
resistance,
objects.
objects
3645,
(1972),
of a set of o b j e c t s
of d i s t i n c t
of
n~0(mod
if
g(n)
in s u c h e x p e r i m e n t s
and
one can more precisely
For
functions
56,
if a n d o n l y
determinant
spectrum,
is t h a t b y w e i g h t i n g
n+1.
9,
and weighted
t e d as a s u m of the m e a s u r e s plans
7, 8,
Schmidt 1978
is h o l d s
functions
9,
maximal
matrices
are
of
3, 5,
(1977,
of o r d e r
5, 6,
(1966),
and Hadene
9.5.
3, 4,
sign
of m e n t i o n e d
s o m e of t h e v a l u e s
g(n)
Cohn
matrix
, a n d the v a l u e s
n = I, 2,
Gilman
the e q u a l i t y
concentrais r e p r e s e n -
idea of w e i g h t e d (not o n e measures.
by
176
For s i m p l i c i t y let us give the w e i g h t i n g problems. we have to weigh
n
objects by
n
of the object and a l l o w a b l e error
weighting; is
n o n - d e p e n d i n g on weight
~ with d i s p e r s i o n
also that there exists an H a d a m a r d m a t r i x of order with two pans)
and
n+1
Suppose that
n
0
2
Suppose
(for w e i g h e r
(for weigher with one pan).
Then there exist w e i g h t i n g methods b a s e d on H a d a m a r d m a t r i x constr u c t i o n a l l o w i n g to reduce the d i s p e r s i o n from case)
and to
no2/(n+1) 2 (for second case).
b e s t p o s s i b l e gains
2
to
02/n
(in first
In a sense they are the
[171].
Note that at p r e s e n t the best known w e i g h t e d plans are b a s e d on H a d a m a r d matrices. T H E O R E M 9.7.
For example,
the f o l l o w i n g t h e o r e m is holds.
[134]. Let there exists an H a d a m a r d m a t r i x of order
n . Then there exist I)
n xp
(p ~ n, p is an a r b i t r a r y natural number)
optimal b a l a n -
ced c h e m i c a l plan; 2) n-1
n-1
objects and for
weightings; 3)
for
D - o p t i m a l b a l a n c e d spasmodic plan for
n-1
A-optimal tendentions
spasmodic plan for
n-1
objects and
weightings.
Proofs of a b o v e - m e n t i o n e d
facts and n o t a t i o n s and more d e t a i l e d
a c q u a i n t a n c e with the w e i g h t e d plans see in h o t e l l i n g (1946), Sloane et al G e r a m i t a et al
(1970,
1976), R a g h a v a r s e
(1976), M c W i l l i a m s et al
9.6. H a d a m a r d m a t r i c e s and rowing. rows of H a d a m a r d m a t r i x of order 8
(1944), Mood
(1971), Banerjee
(1975),
(1979). Gibbs
[2 ] has noted that the
177
1
1
-1
-1
-1
-1
1
1 -1
1 -1
1
1 -1
1 -1
1
1
1
1
hI
1
1
1
1
h2
1
1 -1
-1
h
1
1 -1
-1
h4
H8 = I -I
-I
-I
I
-1
1 -1
I -1
I -I
define nes
the
the
and port ting seats
I
1
I -1
s e a t of r o w e r s
seat of rowers side.
At
trajectory. (see r o w s
I -I
-I
I
I -I
-I
I
h6
1 -1
1 -I
h
I -I
I -I
ho
in a c a d e m i c
such that
such a seat
a n d h5,
oars
the b o a t
Such a winding h7
h
eights.
So,
the
row
h8
alternate
along
starboard
advances
along
slightly
eliminates
respectively).
by
"German"
or
defiside twis-
"Italian"
APPENDIX
UNANSWERED
I. W h e t h e r
for a n y n a t u r a l
a block-circulant mard
matrices
(blocks)
2. H o w m a n y the
same
HI
respectively. 4. L e t
For
m.
and H 2
Hm
Hmn
a given
m
problem
way
der
4n
P~ (H n)
irregular
to state
matrix
that
for a n y
(or g i v e
in t e r m s
?. B e s t
Hadamard
matrices
of o r d e r matrix
number
of
of H a d a -
from
of order
mn.
and
4n
of o r d e r
4mn
matrix
?
of
4)?
matrices Qi
4m
Hadamard
m -= 0 ( m o d
of o r d e r
matrices (Consider
n.
defining
by
analogous
case.) 4n
there
exists
weight
an H a d a m a r d P W / H n)
matrix
of o r -
= I/2 e x c e s s
an contra-example) .
of d e n s i t y [37].
and J.Hammer-
P.Levinston
of w e i g h t
surplus.
Prove
and
+ Pd(n)))
and J.Wallis
that
I I+412+7n~
= 1(I
construct
?
that
n
2( PW(n)
can
c a n be c o n s t r u c t e d
an Hadamard
the c o n d i t i o n s
also Best
problem
one
consisting
high-dimensional
the m i n i m a l
an Hadamard
L e t us g i v e blems
find
n
4mn
m-1 = i=0 E Qi x U mi ' Qi be H a d a m a r d
satisfying = 0
block
exists
for h i g h - d i m e n s i o n a l
6. P r o v e
and
4m ?
be H a d a m a r d there
m
of o r d e r
(matrices)
Is it p o s s i b l e
5. L e t
a unique
be a n
numbers
of o r d e r
blocks
Whether
PROBLEMS
matrix
non-equivalent
Hadamard
3. L e t
order
Hadamard
I
n m 0(mod
8)
'
={
1 ~_]) 7(1 +[ -
-
, n-=4(mod
8}
pro-
179
or
412-V~] n Po (n)
,
J.Hammer,
n/4
weighted
matrix
8)
and
J.Wallis
number,
9.
Let
weighted with
that
and
A square with
(0,-I,+I)
weight
J.Wallis
n ~ 0(mod
matrices
[127].
Proof
that
2 2 2 2, xI > _ x 2 > _ x 3 _> x 4 , n = 4 ( x 1 + x 2 + x 3 + x 4 ~ •
k
4).
W(n,k) . Note
of H a d a m a r d
n 6{12,20,24,28,32,40}
matrix
of order
is d e n o t e d
by
N
W(n,k)
a
provided
[102].
that that
for
every
k,
k ~ n,
for
k=n
this
construction
and
is p r o v e d
U {2k}u
is c a l l e d
n
problems Prove
matrix
and
WW T = kl
Geramita
problem
= I max{2xl,x1+x2+x3+x4}
is a n o d d
DEFINITION.
n ~ 4(mod
R.Levinston
Po (n) where
8)
= { n
8.
, n ~ 0(mod
{3-2k}u
{ 5 . 2 k}
there
problem
exist
coicides
for
, k > 3 [124].
n,
~
~
~
~
0
0
0
0
~
~
~
~
0
0
0
0
~
~
~
~
0
fD
fD
0
I.-h
N
dO
f'D
0
O 0 P'I i'-,I
(I)
H'-
~
~
0
~
~
~
0
~
t~
0
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Vopr. Kibern., Mosk., N 16, part I.
SUBJECT
A-array
4
Abelian
group
Agayan
2
M.A.
A-matrix Aturian
7
2, 4 S.M.
2
autocorrelation - matrix automaton
function 7
theory
Back-circulant
8
matrix
balanced
incomplete
Banerjee
K.S.
Barker
2
block
array
2,
L.D.
0,
Baumert-Hall
type
Baumert-Hall
unities
Bellman
9
Berlecamp
Index
E.R.
is g i v e n
Appendix.
(BIB-design)
1,9
9 3, 4
Baumert-Hall-Goethals-Seidel
R.
design
9
sequence
Baumert-Hall
Baumert
*
6
S.S.
Aizerman
INDEX
method
2
I, 2 Hadamard
matrix
4
3
5, 9
by p a r a g r a p h s ,
0 denotes
Introduction,
AI d e n o t e s
217
B e s t M.R.
6, AI
Bhat
8
V.N.
binary
code
7
block-circulant
Hadamard
matrix
4, AI
block-symmetric
Hadamard
matrix
4
-- g e n e r a l i z e d
Hadamard
-- p a r a m e t r i c block-design geometries
Bose
R.C.
circulant
I
5
P.
8 matrix
theory
circulant
1, 2 6
core
- Abelian
5
group
5
-
generalized
Hadamard
-
generalized
6-code
classic Cohn
Hadamard
J.H.E.
Cooper
J.
0
matrix
5
I, 2 theorem
code
Hadamard
- family
5
4
Hadamard
Cooper-Wallis
cubic
2,
5, 8
matrix
correcting
5
3
matrix
analysis
- design
complex
matrix
9
combinatorial
complete
4
5
F.C.
A.T.
coding
matrix
9
Bussemaker
Cameron
Hadamard
5
0
Bose
Butson
matrix
0, matrix
2 3 6
of W i l l i a m s o n
matrix
6
218
cubic
matrix
Cummings
Data
6
L.
9
processing
Delsarte density
P.
7 5, 9
of H a d a m a r d
matrix
density-of-probability design
theory
diagonal
-
1, 5, 8 7
J.P.
8
- decodable discrete
code
7
orthogonal
- Fourier
function 7
7
D(m,n,k)-sequence
3
D(m,n)-partition D.A.
3
5
Egiasarian Ehlich
7
transformation
- system
Drake
C.O.
H.
6
9
eigenvalue
7,
eigenvector entropy
7
2
set
matrix
Dillon
function
8
matrix
difference
6
9
7 7
equivalent
Hadamard
Euclidean
coordinates
- space
matrix I
7
extended
code
5
extremal
geometrical
constants
I
219
Factorable fast
Hadamard
algorithm
filtration finite
matrix
8,
9
7
geometrics
0
- projective F-matrix Fourier
plane
sum
Generalized
5, 7
Williamson
matrix
- Hadamard
matrix
k-elemental
on groups
Williamson
- Yang matrix A.V.
3,
Goethals
J.M.
0, 5,
Goethals-Seidel M.D.E
approach
S.W.
2
Gordon
B.
I
Good matrix
7
9 I, 2,
3, 4
theory
8
group
theory
0, 8
function
3 sequences
graph
2
7
matrix
- array
matrix
0
supplementary
Hadamard
5
9, AI
array
Golomb
Haar
5
5
Geramita
Colay
2
5
hyperframe
- parametric
Golay-Turyn
matrix
3
Hadamard
Golay
9
0
- 6-code
-
8,
2
- matrix
-
5
2,
0, 2, 4, 4
7, 8,
9, AI
220
- function
7
- problem
0
- product
I
system
-
4
- transformation Hammer
A.
0
Hartley
H.C.
Hedeyat
A.
Hermitian
8 0
matrix
7
- function hybrid
7
orthogonal
base
high-dimensional classic
-
Hadamard
Hadamard
- generalized - improper
Hadamard
Image
incidence
-
John
6
3 matrix
theory
0,
P.W.M.
cubic
8
4
7
0
equivalence
irregular
6
7
compressing
integral
matrix
Hadamard
information
6
6
matrix
incomplete
6
9
processing
- coding
matrix
7
H.
Hadamard
6, 6
matrix
matrix
space
Hotelling
Hadamard
design
- Williamson Hilbert
matrix
matrix
Hadamard
- orthogonal - proper
7
I matrix
AI
221
Johnson
E.C.
Jungnical
8
D.
5
Karhunen-Loeve
decomposition
- filter
7
Kasami
T.
7
Kiefer
J.
8
Kirton
H.C.
8
Khachatrian
G.G.
Kotelnikov Kozirev
theorem
V.P.
Kronecker
Lagrange latin
matrix
4
theorem
2
1-elemental
0,
3,
L-distance
Levinston linear
7 V.I. P.
- code
7,
(van
AI
Lint
J)
8 7
9
McWilliams
Markova
0
function
R.
Markov
9
7
Lipshits Lynch
3
algebra
J.
I, 4
4
semi-frame
Levenstein
8
hyperframe
frame
-
Lint
7
9
squares
-
7
F.J.
signals E.V.
Matevosian
A.K.
7,
9
2,
7
7 8
222
maximal
code
maximum
determinant
Milas
J.
7 problem
H.
monomial
permutation
6
m-parallelipiped m-space
channel
multiplicative
non-periodic normalized
0,
Hadamard
C.W.
function
matrix
0-code
theory
]
3 ~-code
3
0
linear
detection
- balanced
chemical
0 plan
7
- linear
filtration
- Wiener
filter design
- array base
matrix
8
I
- generalized
code
7,
autocorrelation
n-symbolical
-
Hadamard
I coding
orthogonal
7
5
generalized
M.
Optimal
0,
group
N-dimensional
noiseless
2
7
multiple-access
number
I
I
A.G.
R.C.
Norman
matrix
I
Muchopadhyay
Newman
9
I
Miyamoto
Mullin
0,
7 7
I,
4
5 7
-Chebyshev-Hadamard
system
3
223
-
-
F-square
design
Hadamard
transformation
- table
0
- F-square
configuration
system
-
0
0
- transformation
Palay
7
5
matrix
-
8
R.E.A.C.
0
0,
4,
7
Palay-Wallis-Whiteman
method
2
parametric
family
2
Williamson
Hadamard
-
matrix
- williamson Yang
matrix
matrix
recognition
partial
factor S.E.
7
design
M.
Plotkin
hypothesis
0,
- array
3,
2
- method
7 3
- partition
3
- theorem
4
autocorrelation E.C.
projective
function
7 plane
Quasi-symmetrix
Radon
7 0
- boundaries
Posner
8
9
Plotkin
periodic
2
2
pattern
Payne
4
function
8
design
I
8
224
Raghavarao
D.
Rao
5
K.R.
rapid
(fast)
9
algorithm
- Hadamard
transformation
Read-Maller
code
rectangular
matrix
regular
graph
7
matrix
relation
reverse
8
7
transformation
Robinson
P.J.
3
Rutledge
W.A.
1
Ryser
I, 4
5
- Hadamard Relay
7
H.J.
0, 8
Sarukhanian
A.G.
Scarpis
U.
Schmidt
K.W.
Seberry
J.
5, 8
Seidel
J.J.
0,
Seiden
E.
3
9
1, 9
8 3
C.E.
Shlichta
2,
0, 4
semi-partition Shennon
7
7
problem
6
Shrikhande
S.S.
5, 6, 8
Sidelnikov
V.M.
0
Singhi
N.M.
8
skew-symmetric Slepian Sloane
D. N.J.
S-matrices spatial
matrix
I
7 7, 9 I, 4
generalized
Hadamard
matrix
225
- Hadamard special
Hadamard
spectral
packing
9
system
Stanton
R.G.
8 8
J.J.
T.
I
8,
9
story-by-story
Kronecker
Street
5, 8
P.
strongly
0,
regular
supplementary -
-
matrix
graph
Golay
m-sequences
sequences 3
of H a d a m a r d
Sylvester
J.J.
symmetric
incomplete
- BIB
I
- hyperframe
2
G.
matrix
8
telemetric
system
t-design
8
T-matrix
0, 2,
Trachtman
A.M.
T-sequences
3 5
Turyn
sequences
Turyn
R. code
7
3
0,
6
block-design
2, 8
Y.
T-user
3
0, 4
design
Szekers
4,7
0, 9
Q(n,t)-sequences
surplus
Taki
6
2, 8
Stainer
Storer
matrix 7
E.
Stiffler
6
analysis
spherical Spence
matrix
3
2, 3, 8, 7
9
(SBIB)
226
-
uniquely
Uniquely unit
decodable
decodable
matrix
T-user
matrix
5,
Vilenkin-Kronecker
Wallis
array
Wallis
J.
Wallis
W.D.
0,
2,
array system
equivalence
Weldon
E.S.
Williamson
0,
array
2,
3
7
2,
8
I,
2,
2,
- method
9
4,
3 6
2
- theorem
2 J.
0,
- family
2,
6,
3,
Hadamard
filter
C.H.
Yang
matrices
I,
matrix
0 6, 2,
2,
6
9
6
7
S.V.
Yang
array
I, 4
I,
- matrices
-
9
7
A.L.
Yablonskiy
8,
function
weight
Wiener
6,
5
Walsh-Hadamard
type
3,
8
- matrix
-
6
system
I,
function
Williamson
code
I
Wallis-Whiteman
Whiteman
code
I
Vandermonde
Walsh
basic
9 5
4
7
227
- theorem Youden
Zinovjev
design
V.A.
5 8
0,
9
