Hadamard Matrices and Their Applications

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 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_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 ( 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)
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)


_ 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

c~ fD

0

©

m

Q

H

O

rD ct H

m

0 Q

Q

~

N

I

0

Co

0

t~

0

cn

f

~D

Z U H

®

o

~

~

~

~

~

~

~

~

:

o

~

~

~

~

~

~

~

0

~

~

0

0

~

~

it

~

~

o

~

~

~

o

~

~

o

~

~

0

~

~

m

~

~

~

~

~

~

~

~

~

~

~

~

~

o

~

~

0

o

~

~

~

~

X

~

~

~

~

~

~

~

~

n

rO

~

~

~

~

~

~

~

~

~

~

~

~

~

~

0

~

~

~

t

0

0

0

0

0

0

0

0

~

~

0

0

0

0

0

0

0

O

O

~

~

~

~

~

~

~

~

~

it

~

~

~

~

~

~

~

~

o

~

~

-

Z

~

~

o

~

~

~

~

~

®

~

~

~

~

~

o

o

o

~

~

~

~

~

~ 5 ~

~

~

o

~

~

~

o

~

o

~

~

~

~

-

~

o

o

~

-

~

~

~

~

~

~

~

o

~

~

~

~

~

~ o

~

~

o

~

~

5

~

~

~

~

~

~

~

~

~

Z

~

~

~

~

~

~

~

~

Z

o

S

o

~

~

~

~

~

~

~

~

~

5

~

~

~

~ t~

~

~

~

0

~

~

~

~

~

0

0

~

~

~

~

~

0

o

~

~

~

0

~

~

Ii

0

~

~

~

0

0

0

~

~

~

~

8~4=

~

0

186

187

n

~

~

0

~

~

~

0

~

~

188

~

~

0

~

~

~

0

~

~

~

0

~

~

~

0

~

~

~

0

~

~

~

0

~

~

~

0

~

~

~

0

~

~

~

0

~

~

~

0

~

~

~

0

~

~

~

0

~

189

ii

190

ii

191

A

o r~ -;-I ..~ 0 U

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132-142,

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Linear

197-207.

On H a d a m a r d

matrices.

J.Comb.

Theory,

Set. A,

matrices.

J.Comb.

149-164.

Wallis

Waliis

J.

(1976).

Ser. A, J.

Wallis

Wallis

On the e x i s t e n c e

21,

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188-195.

A computer

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147-154.

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Math.,

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Multilinear 295

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119-125.

J.

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294

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Ser. A,

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293.

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4,

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289. W a l l i s

292.

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(1972b).

Aust.Conf.

291.

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5, 203-204.

8, 242-257.

288. W a l l i s

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Some c l a s s e s

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of H a d a m a r d

matri-

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matrices.

213

Bull.Aust.Math.Soc. 300

301

Wallis W.D.

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incomplete block designs and graphs.

B u l l . A u s t . M a t h . Soc.

Wallis W.D.

t h e o r e m for

(1970a). A n o n - e x i s t e n c e

Aust.Math.Soc., 302

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304

Ser. A,

1,425-430.

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11, 381-383.

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designs. J.Comb.

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Isr.J.Math°

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103-106, Tunra, Newcastle,

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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