Hadamard matrices, orthogonal designs and construction algorithms

TLDR: In this paper, the authors discuss algorithms for the construction of Hadamard matrices and give algorithms for constructing orthogonal designs, short amicable and amicable sets for use in the Kharaghani array.

Abstract: We discuss algorithms for the construction of Hadamard matrices. We include discussion of construction using Williamson matrices, Legendre pairs and the discret Fourier transform and the two circulants construction.Next we move to algorithms to determine the equivalence of Hadamard matrices using the profile and projections of Hadamard matrices. A summary is then given which considers inequivalence of Hadamard matrices of orders up to 44.The final two sections give algorithms for constructing orthogonal designs, short amicable and amicable sets for use in the Kharaghani array.

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University of Wollongong University of Wollongong
Research Online Research Online
Faculty of Informatics - Papers (Archive)
Faculty of Engineering and Information
Sciences
January 2002
Hadamard matrices, orthogonal designs and construction algorithms Hadamard matrices, orthogonal designs and construction algorithms
S. Georgiou
National Technical University of Athens, Greece
C. Koukouvinos
National Technical University of Athens, Greece
Jennifer Seberry
University of Wollongong
, jennie@uow.edu.au
Follow this and additional works at: https://ro.uow.edu.au/infopapers
Part of the Physical Sciences and Mathematics Commons
Recommended Citation Recommended Citation
Georgiou, S.; Koukouvinos, C.; and Seberry, Jennifer: Hadamard matrices, orthogonal designs and
construction algorithms 2002.
https://ro.uow.edu.au/infopapers/308
Research Online is the open access institutional repository for the University of Wollongong. For further information
contact the UOW Library: research-pubs@uow.edu.au

Hadamard matrices, orthogonal designs and construction algorithms Hadamard matrices, orthogonal designs and construction algorithms
Abstract Abstract
We discuss algorithms for the construction of Hadamard matrices. We include discussion of construction
using Williamson matrices, Legendre pairs and the discret Fourier transform and the two circulants
construction. Next we move to algorithms to determine the equivalence of Hadamard matrices using the
pro?le and projections of Hadamard matrices. A summary is then given which considers inequivalence of
Hadamard matrices of orders up to 44. The ?nal two sections give algorithms for constructing orthogonal
designs, short amicable and amicable sets for use in the Kharaghani array.
Disciplines Disciplines
Physical Sciences and Mathematics
Publication Details Publication Details
This book chapter was originally published as Georgiou, S, Koukouvinos, C and Seberry, J, Hadamard
matrices, orthogonal designs and construction algorithm, in Wallis, WD (ed), Designs 2002: Further
Combinatorial and Constructive Design Theory, Kluwer Academic Publishers, Norwell, Massachusetts,
2002, 133-205. Original book available here.
This book chapter is available at Research Online: https://ro.uow.edu.au/infopapers/308

Hadamard matries, orthogonal designs and
onstrution algorithms
S. Georgiou, C. Koukouvinos
Department of Mathematis
National Tehnial University of Athens
Zografou 15773, Athens, Greee
and
Jennifer Seb erry
Sho ol of IT and Computer Siene
University of Wollongong
Wollongong, NSW 2522, Australia
Contents
1 Algorithms for onstruting Hadamard matries 2
1.1 Hadamard matries onstruted from Williamson matries . . . . . . . . . . . . 2
1.1.1 Results from previous searhes . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.2 Searh metho d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.3 Searh results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Hadamard matries from Williamson matries for non prime orders . . . . . . . . 10
1.2.1 The metho d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.2 The algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 Hadamard matries from generalized Legendre pairs using the disrete Fourier
transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.1 Denitions and notations . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.2 Some preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.3 Legendre sequenes and mo died Legendre sequenes . . . . . . . . . . . 15
1.3.4 The PSD test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3.5 Empirial performane of the PSD test for binary sequenes . . . . . . . . 17
1.4 Hadamard matries from generalized Legendre pairs using supplementary dier-
ene sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4.1 Some preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4.2 Twin prime power onstrution . . . . . . . . . . . . . . . . . . . . . . . . 19
1.4.3 The algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.5 Hadamard matries onstruted from two irulant matries . . . . . . . . . . . . 22
2 On inequivalent Hadamard matries 24
2.1 Basi denitions and preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2 The prole riterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1

2.3 The pro jetion and Hamming distane distribution algorithms . . . . . . . . . . . 25
2.4 Appliation of the new riterion to Hadamard matries of small orders . . . . . . 29
2.4.1 Hadamard matries of order
n
= 4
;
8
;
12 . . . . . . . . . . . . . . . . . . . 30
2.4.2 Hadamard matries of order
n
= 16 . . . . . . . . . . . . . . . . . . . . . 30
2.4.3 Hadamard matries of order
n
= 20 . . . . . . . . . . . . . . . . . . . . . 30
2.5 Inequivalent Hadamard matries . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5.1 Hadamard matries of order
n
= 24 . . . . . . . . . . . . . . . . . . . . . 30
2.5.2 Hadamard matries of order
n
= 28 . . . . . . . . . . . . . . . . . . . . . 31
2.5.3 Hadamard matries of order 32 . . . . . . . . . . . . . . . . . . . . . . . . 32
2.5.4 Hadamard matries of order 36 . . . . . . . . . . . . . . . . . . . . . . . . 32
2.5.5 Hadamard matries of order 40 . . . . . . . . . . . . . . . . . . . . . . . . 32
2.5.6 Hadamard matries of order 44 . . . . . . . . . . . . . . . . . . . . . . . . 32
3 Algorithms for onstruting orthogonal designs 32
3.1 Basi denitions and preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2 Constrution algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.1 The matrix based algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.2 The extension algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2.3 The merge algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 Amiable sets of matries and onstrutions of orthogonal designs using the
Kharaghani array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4 Short amiable sets and Kharaghani type orthogonal designs 46
4.1 Preliminary results and basi denitions . . . . . . . . . . . . . . . . . . . . . . . 46
4.2 Construtions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.3 Some general results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Abstrat
We disuss algorithms for the onstrution of Hadamard matries. We inlude disussion
of onstrution using Williamson matries, Legendre pairs and the disret Fourier transform
and the two irulants onstrution.
Next we move to algorithms to determine the equivalene of Hadamard matries using
the prole and pro jetions of Hadamard matries. A summary is then given whih onsiders
inequivalene of Hadamard matries of orders up to 44.
The nal two setions give algorithms for onstruting orthogonal designs, short amiable
and amiable sets for use in the Kharaghani array.
1 Algorithms for onstruting Hadamard matries
1.1 Hadamard matries onstruted from Williamson matries
An Hadamard matrix
H
of order
n
has elements

1 and satises
H H
T
=
nI
n
. These matries
are used extensively in o ding and ommuniations (see Seb erry and Yamada [90℄). The order
of an Hadamard matrix is 1, 2 or
n

(0 mo d 4). The rst unsolved ase is order 428. We
use Williamson's onstrution as the basis of our algorithm to onstrut a distributed omputer
searh for new Hadamard matries. We briey desrib e the theory of Williamson's onstru-
tion below. Previous omputer searhes for Hadamard matries using Williamson's ondition
2

are desrib ed in Setion 1.1.1. The implementation of the searh algorithm is presented in
Setion 1.1.2, and the results of the searh are desrib ed in Setion 1.1.3.
Theorem 1 (Williamson [104℄)
Suppose there exist four
(1
;

1)
matries
A
,
B
,
C
,
D
of
order
n
whih satisfy
X Y
T
=
Y X
T
; X; Y
2 f
A; B ; C ; D
g
Further, suppose
AA
T
+
B B
T
+
C C
T
+
D D
T
= 4
nI
n
(1)
Then
H
=
2
6
6
6
4
A B C D

B A

D C

C D A

B

D

C B A
3
7
7
7
5
(2)
is an Hadamard matrix of order
4
n
onstruted from a Wil liamson array.
Let the matrix
T
given b elow b e alled the shift matrix:
T
=
2
6
6
6
6
6
4
0 1 0
  
0
0 0 1
  
0
     
:
0 0 0
  
1
1 0 0
  
0
3
7
7
7
7
7
5
(3)
and note
T
n
=
I ;
(
T
i
)
T
=
T
n

i
(4)
If
n
is o dd,
T
is the matrix representation of the
n
th ro ot of unity
!
,
!
n
= 1.
Let
8
>
>
>
<
>
>
>
:
A
=
P
n

1
i
=0
a
i
T
i
; a
i
=

1
; a
n

i
=
a
i
B
=
P
n

1
i
=0
b
i
T
i
; b
i
=

1
; b
n

i
=
b
i
C
=
P
n

1
i
=0

i
T
i
; 
i
=

1
; 
n

i
=

i
D
=
P
n

1
i
=0
d
i
T
i
; d
i
=

1
; d
n

i
=
d
i
(5)
Then matries
A; B ; C ; D
may b e represented as p olynomials. The requirement that
x
n

i
=
x
i
; x
2 f
a; b; ; d
g
fores the matries
A; B ; C ; D
to b e symmetri.
Sine
A; B ; C ; D
are symmetri, (1) b eomes:
A
2
+
B
2
+
C
2
+
D
2
= 4
nI
n
and the relation
X Y
T
=
Y X
T
b eomes
X Y
=
Y X
whih is true for p olynomials.
Denition 1
Wil liamson matries are
(1
;

1)
symmetri irulant matries. As a onsequene
of being symmetri and irulant they ommute in pairs.
We use the following theorem of Williamson's as the motivator for our searh algorithm:
3

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Q4. how many k multipli ations do a hamming distan e take?

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