ON ORTHOGONAL MATRICES

ON ORTHOGONAL MATRICES MAJID BEHBAHANI B a c h e l o r of Engineering, Shahid B e h e s h t i University, 2002 A Thesis Submitted to the School of G...
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ON ORTHOGONAL MATRICES

MAJID BEHBAHANI B a c h e l o r of Engineering, Shahid B e h e s h t i University, 2002

A Thesis Submitted to the School of Graduate Studies of the University of Lethbridge in Partial Fulfilment of the Requirements for the Degree M A S T E R OF S C I E N C E

Department of Mathematics and Computer Science University of Lethbridge LETHBRIDGE, ALBERTA, CANADA

©Majid Behbahani, 2004

ABSTRACT On Orthogonal Matrices Majid Behbahani D e p a r t m e n t of M a t h e m a t i c s and C o m p u t e r Science U n i v e r s i t y of Lethbridge M . Sc. T h e s i s , 2004 Our main aim in this thesis is to study and search for orthogonal matrices which have a certain kind of block structure. The most desirable class of matrices for our purpose are orthogonal designs constructible from 16 circulant matrices. In studying these ma­ trices, we show that the

1,1,1, 9) is the only orthogonal design constructible

OD(l2;

from 16 circulant matrices of type OD(4n; 1,1, l , 4 n — 3), whenever n > 1 is an odd integer. We then use an exhaustive search to show that the only orthogonal design con­ structible from 16 circulant matrices of order 12 on 4 variables is the

OD(12;

1,1,1, 9).

It is known t h a t by using of T-matrices and orthogonal designs constructible from 16 circulant matrices one can produce an infinite family of orthogonal designs. To com­ plement our studies we reproduce an important recent construction of T-matrices by Xia and Xia. We then turn our attention to the applications of orthogonal matrices. In some recent works productive regular Hadamard matrices are used to construct many new infinite families of symmetric designs. We show that for each integer n for which 4n is 2

the order of a Hadamard matrix and 8 n — 1 is a prime, there is a productive regular 2

2

2

Hadamard matrix of order 1 6 n ( 8 n — l ) . As a corollary, we get many new infinite 2

2

classes of symmetric designs whenever either of 4n(8n — 1) — 1, 4 n ( 8 n — 1) + 1 is a prime power. We also review some other constructions of productive regular Hadamard matrices which are related to our work.

To My

Parents

ii

ACKNOWLEDGEMENTS

First I would like to thank my family for always supporting me. I would like to extend my warmest gratitude to my supervisor, Dr. Hadi Kharaghani, for his immense amount of support, advice, and guidance throughout the process. I couldn't have done any of this without his support. Acknowledgement also needs to be extended to my co-supervisor, Dr. Wolf Holzmann, for his constant help and continued support. My special thanks also to the members of my defence committee, Dr. Saurya Das, Dr. Wolf Holzmann, Dr. Yury J. Ionin, Dr. Hadi Kharaghani, and Dr. Mark Walton, for their valuable comments, suggestions, and corrections. I am grateful to Dr. Amir Akbary for his helpful conversations and valuable sug­ gestions. His help is greatly appreciated. I also acknowledge Dr. Hadi Kharaghani as well as the university of Lethbridge for supporting me financially. I am grateful to Sean Legge and Stephen Ney for doing the proofreading and pro­ viding helpful suggestions.

iii

Contents 1

I n t r o d u c t i o n and s t a t e m e n t s of results

1

2

O r t h o g o n a l matrices

6

2.1

Orthogonal designs

6

2.2

An exhaustive search

13

2.3

A non-existence theorem

15

2.4

A family of T-matrices

16

3

P r o d u c t i v e Regular H a d a m a r d M a t r i c e s and S y m m e t r i c D e s i g n s

33

3.1

Introduction

33

3.2

Bush-type Hadamard matrices

38

3.3

The Kronecker product of Bush-type Hadamard matrices and produc­ tive regular Hadamard matrices

4

42

A N e w Class of P r o d u c t i v e R e g u l a r H a d a m a r d M a t r i c e s

46

4.1

Introduction

46

4.2

A regular class of Hadamard matrices

47

4.3

A productive class of regular Hadamard matrices

50

iv

Chapter 1 Introduction and statements of results Definition 1.1 l

such that:

HH

A Hadamard = nl , n

matrix

where I

H

of order

n is an n

is the identity

n

matrix

x

n matrix

of order

±1

with

entries

n.

The order of a Hadamard matrix must be 1, 2 or a multiple of 4. The conjecture that all Hadamard matrices of order 4n exist for every positive integer n is still an important open problem. One way to generalise Hadamard matrices is by means of orthogonal designs. Definition 1.2 is a positive {0, ± X i , ± x

An

orthogonal

integer, t

}

denoted

design OD(n;

(the Xi commuting

A of order u \ , u ) ,

n and type u\, ...,u , t

is an n x n matrix

t

indeterminates)

with

where

each Ui

entries

from

satisfying

Despite extensive work on the existence and properties of orthogonal designs, not many significant results are known about these matrices. One very useful constructive method for orthogonal designs is by means of T-matrices. Definition 1.3 matrices

Four

if they satisfy

type-1

{0, ± 1 }

the following

matrices

T\,

conditions:

1

T

2)

T

3;

and

T

4

of order

n are

T-

1. Ti DTj — 0 where i ^ j ; 2- Eli \n = J; By combining orthogonal designs constructible from 16 circulant matrices with T matrices one can construct a large family of very useful orthogonal designs. Although there is an orthogonal design of order 20 constructible from 16 circulant matrices, nothing is known about the existence of such matrices of order 12. In chapter 2 we

OD(12; 1,1,1, 9) and OD(A; 1,1,1,1) are the only orthogonal designs of type OD(An; 1,1,1, An — 3) constructible from 16 circulant matrices when n is odd. We also use an exhaustive search to show that OD(12; 1,1,1, 9) is the only orthogonal will show t h a t

design of order 12 on 4 variables constructible from 16 circulant matrices. The existence of amicable set of T-matrices has proven to be instrumental in the construction of orthogonal designs.

The T-matrices Ti, T , T , andT^ of order n are amicable T-matrices if they satisfy the amicability condition:

Definition 1.4

2

3

T{T\ - T T{ + T{T\ - T{P% = 0. 2

(Note that there is no specific order for TjS, and we can rename them to satisfy th amicability condition in this order.) In part of chapter 2 we show that amicable T-matrices of

odd order do

not exist. We

then conclude the chapter by a very important recent result concerning finite fields [17]. Our hope is to develop and use this result in the future to produce some positive results on the existence of T-matrices. We devote the remaining chapters to the applications of Hadamard matrices in the construction of symmetric designs.

A symmetric (v,k, A)-design is an incidence system (P, B) in which P = {pi,P2, • • • ,Pv} is a set of v points and B = {bi,... ,b } is a set of v blocks, each block being a k-subset of P such that any two points of P are incident with exactly blocks of B. Definition 1.5

v

2

Symmetric designs can be expressed by their incidence matrices. Definition 1.6 A — [dij]

such

The incidence

matrix

of a symmetric

(v, k, \)-design

is avxv

matrix

that

J 1 I

if Pi E bj otherwise.

0

A (0, l ) - m a t r i x A is an incidence matrix of a symmetric (v, k, A) design if and only if = (jfc - A)/ + A J.

AA*

In this thesis we only study symmetric designs constructed from productive regular Hadamard matrices. The class of productive Hadamard matrices was denned by Yury Ionin in [4]. A regular Hadamard matrix is a Hadamard matrix with constant row sum. Definition 1.7

A regular

is a set H of matrices is a bijection, 1.

with

such

Hadamard

matrix

H with

row sum 2h and a cyclic

row sum 2h is group

G =
where 5 : H —> Ti

that

HeH;

2. For

any H

l

:

H



2

H,

t

(SH )(SH ) l

=

2

H H\; X

3. \G\ = 4 | / i | ; 4- T,aeG

aH

=

2

J

W\ -

Productive Hadamard matrices are normally used in Balanced Generalised Weighing matrices over cyclic groups. Definition 1.8

Let

weighing matrix

BGW(v,k,X)

such that that for X/\G\

G

be a multiplicatively is a matrix

each row and each column any h

copies

i, the multiset of every

element

written

ofW 1

{w^jW^

W

group.

of order

— [wij]

contains

exactly

: 1 < j < v, w j h

ofG.

3

A

balanced generalised v with

k non-zero

Wij



entries

^ 0, Wij ^ 0} contains

G and

U {0} such exactly

A large class of balanced generalised weighing matrices of the type ,m—l

m

BGW((q -l)/(q-l),q>

over a cyclic group G, where q is a prime power, m is a positive integer and the order of G divides 5 — 1, is known t o exist. A classical construction due to Ionin is as follows: T h e o r e m 1.9 / / there and

if q =

symmetric

(2h — l ) design

2

is a productive is a prime

with

regular

power

Hadamard

then

for

matrix

H with

any positive

row sum 2h

integer

m there

is a

parameters:

Bush-type Hadamard matrices are all known to be productive. Definition 1.10 blocks

1
fe=0 Define EI — EJ AS I = j ( m o d 8m + 8), SI = SJ, TI = TJ as i = j ( m o d 4m + 4). It is easy t o see that GEI = E I,

GSI = S I,

I+

I+

$o = $I = A{E ,EI),

I + I

.

Define

AE , 0

I = 1 , 2 , . . . , 8 m + 7,

0

$i =

and G% — T

as i = j ( m o d 8m + 8).

We have 2m AEI

=

2m

J2/Z9

8 ( M + 1 ) J + I E

J=0 J'=0

2m 2m J=0 J'=0 2m

J=0

=

2m /=0

3**o

18

9

8 I M + 1 ) J

'

+ I

for i — 0 , 1 , . . . , 8m + 8. Similarly we can show that i

A(Ei, Ej) — g ^j-i for i ^ j, and also $i = g^-i

=

g^sm+s-i

for i — 1, 2 , . . . , 8m + 7. 2

Let G = GF(q ) be an extension of GF(q) then

L e m m a 2.12

E \JE 0

= GF( y.

4m+4

q

q

l

GF(q). Consider the polynomial x ~ Q 1 — 0 in GF(q ). Clearly this polynomial has q — 1 roots, since the order of the multiplicative group of GF(q) is q — 1 we have x ~ = 1 for all x £ GF(q). Thus all the roots of the polynomial x ~ 0 1 = 0 are in GF(q). On the other hand we have ( 4(m+i)),-i *-i 1 = 1 e 1 - 0. 4

Proof

It is sufficient to show that g (

m + 1

) £

2

q

q

l

5

4

Since g (

m + 1

l

e

:

=

gq

e

) is one of the roots of the polynomial

q

l

x~

© 1 = 0, it is in

GF(q).

By a

simple counting we have:

E UE 0

= GF( y.

4m+4

q

• Here we need to show that:

[1] The set of all non-zero squares in GF(q) form a (q, \(q — difference set.

L e m m a 2.13

Proof

Let a be a primitive element of GF(q) and let D be the set of all non-zero

squares in

GF(q).

We have

0

D = {a , a ,...,

3

—1 ^ D and — D = {a, a , . . . , a

4 m + 1

}.

2

a

4 m

} . Since

Let a* and 2bi

1 = o > ' Ga . 2t

£ D we have 2t

2

+t)

a = a^

19

0 a

2 ( i

'

+ t )

4) we have

i — 1 , . . . n be the set of all

integers such that

Then for any a

q = 3(mod

.

1), \(q

— 3))-

So every representation of 1 as the difference of the elements of D gives us a represen­ 2t

tation of a 2t+1

a

as the difference of the elements of D and vice versa. For any element 2t+1

£ D we have a

€ — D so 2t+l

a

2t+1

a =

2t

= —a

2(b t') Q

a

i+

A

for some integer

we have

2(2 + ')_ I

T

Thus every representation of 1 as the difference of the elements of D gives us a rep­ 2t+1

a

resentation of

difference set in

as the difference of the elements of

GF(q).

Clearly

k — \D\ =

D

and vice versa. So

|(

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