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Hadamard Matrices and Their Applications

Hadamard Matrices and Their Applications

by K. J. Horadam


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ISBN-13: 9780691119212
Publisher: Princeton University Press
Publication date: 12/03/2006
Edition description: New Edition
Pages: 280
Product dimensions: 6.00(w) x 9.25(h) x (d)

About the Author

K. J. Horadam is Professor of Mathematics and leads the Information Theory and Security Research Group at RMIT University, Melbourne, Australia.

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Hadamard Matrices and Their Applications

Chapter One


The purpose of this book is three-fold: to report the current status of existence and construction problems for Hadamard matrices and their generalisations; to give an accessible account of the new unifying approach to these problems using group cohomology; and to support an understanding of how these ideas are applied in digital communications. I have tried to present results and open problems with sufficient rigour, and direction to the literature, to enable readers to begin their own research, but with enough perspective for them to gain an overview without needing in-depth knowledge of the algebraic background.

The book has two Parts. In Part 1, consisting of four Chapters, our present understanding of Hadamard matrices, generalised Hadamard matrices and higher dimensional Hadamard matrices is summarised. One Chapter is devoted to introduction and explanation of the main applications of Hadamard matrices in digital signal and data sequence processing, principally for spectral analysis and signal error protection, separation or encryption.

Generalised Hadamard matrices and higher dimensional Hadamard matrices are each natural enlargements of the class of Hadamard matrices, in the direction ofentries not restricted to {±1} and not restricted to 2-dimensional (2-D) arrays, respectively. Part 1 contains the basic definitions and properties of these three types of Hadamard matrices and, for each of them, a status report on recent results using classical techniques. The two ideas from which Warwick de Launey and I developed the group extensions approach to Hadamard matrices: group development of Hadamard matrices and construction of higher dimensional Hadamard matrices from relative difference sets are highlighted.

Part 2, also consisting of four Chapters, develops in detail the unifying group extensions approach to existence and construction of the three types of Hadamard matrices covered in Part 1. Some necessary algebraic background is included. This Part covers the major theoretical advances made over the past 15 years, culminating in the Five-fold Constellation, which identifies cocyclic generalised Hadamard matrices with particular 'stars' in four other areas of mathematics and engineering: group cohomology (factor pairs), incidence structures (divisible designs), combinatorics (relative difference sets) and signal correlation (perfect arrays). The work in this Part has not been collected before, or is accessible only in journal articles. Some is not yet published.

The latter half of Part 2 introduces less mature, but very exciting, theoretical results on the atomic structure of cohomology classes. These shift orbits have remained invisible for nearly a century, but carry the statistical information about distributions of the entries of cocyclic matrices that determines whether or not 2 they will produce Hadamard matrices, high-distance error-correcting codes and low-correlation sequences. Finally, the first applications of the theory of cocyclic Hadamard matrices to multiphase signal and data sequence processing are presented. We construct novel and optimal families of such cocyclic generalised Hadamard matrices and their corresponding Generalised Hadamard Transforms, codes and sequences.

Half the open research problems arise in this last quarter of the book.

A summary of each Chapter follows.

Chapter 2 covers basic definitions and properties of Hadamard matrices, in abbreviated form. There are many excellent texts [288, 1, 123, 315], reviews [68, 69] and databases [212, 287, 297], describing Hadamard matrices and their numerous constructions in more detail; the intention here is to provide a succinct summary and update of research over the past decade or so. Direct constructions of Hadamard matrices by Sylvester, Paley and Williamson and from Hadamard designs are described and illustrated.

More modern techniques of constructing Hadamard matrices, by patterning entries according to the multiplication table of a group, are treated next. This is our first link to cocycles and cocyclic Hadamard matrices. In the final section of Chapter 2, advances towards direct confirmation of the celebrated Hadamard Conjecture, and improved asymptotic support for it, are outlined, as is progress on the circulant Hadamard conjecture.

The purely intellectual excitement and challenge of finding new Hadamard matrices and homing in on confirmation of the Hadamard Conjecture is heightened by the knowledge that they are marvellously useful. Chapter 3 is devoted to two of their three principal applications: Hadamard transform spectroscopy and object recognition, and coding of digital signals. Applications in design of experiments are not included. Most emphasis is placed on coding of digital signals or data sequences for error correction, separation, correlation or encryption.

Each application area is introduced briefly to explain how the Hadamard matrix is applied, but in enough detail, and in the language of the application, to explain current trends. My aim is to bridge the two worlds: to translate the physical application into terms a pure mathematician will appreciate and the theoretical structure into terms an applied mathematician, computer scientist or communications engineer can adapt and use.

Chapter 4 moves us from Hadamard matrices to generalisations where matrix entries are not restricted to {±1}. More than one direction for enlargement of the class of Hadamard matrices has flourished, but generalisations to maximal determinant matrices, weighing matrices, orthogonal designs and nonsquare matrices will not be covered. The two main formulations we treat are complex Hadamard matrices (invertible, with entries on the complex unit circle) - especially those with entries which are roots of unity, called Butson matrices here - and generalised Hadamard matrices (with entries from a finite group N, for which the inner quotient of any distinct pair of rows in the integral group ring [??]N equals [lambda] ([[summation].sub.u]member of] N] u), for some fixed integer [lambda]). To complicate matters, in the literature the term complex Hadamard matrix often refers only to a Butson matrix with entries in {±1, ± [square root of -1]}, of which those with uniformly distributed rows are also called quaternary generalised Hadamard matrices. Although complex Hadamard matrices will be revisited on occasion, the principal subject of this book is generalised Hadamard matrices. Jungnickel's seminal 1982 result, relating generalised Hadamard matrices, class regular divisible designs and relative difference sets, underscores the richness of the interconnections between these areas and the group extensions approach described in the second part.

This Chapter follows the structure of Chapter 2, for each of Butson, complex Hadamard and generalised Hadamard matrices in turn, illustrated with numerous examples. One section covers their applications to multiphase signals and sequences. The final section is new work, unifying the two formulations in the invertible Generalised Butson Hadamard matrices, which include all complex Hadamard matrices and all invertible generalised Hadamard matrices, and their Generalised Hadamard Transforms.

Chapter 5 enlarges the class of Hadamard matrices from 2-D to n-dimensional arrays with entries from {±1}. It deals with n-dimensional proper Hadamard matrices, introduced by Shlichta in 1971, which have the property that all 2-D sub-arrays obtained by fixing any n - 2 coordinates are Hadamard matrices.

Despite a strong presumption of their utility - based on that of Hadamard matrices - and their formative role in development of the group extensions approach to Hadamard matrices, remarkably little is known about higher dimensional proper Hadamard matrices. The first monograph on the subject is Yang [334]. A summary of construction techniques, relationships between these techniques, equivalence classes and applications to Boolean functions useful for cryptography and to error-correcting array codes is presented.

Higher dimensional proper Hadamard matrices were central to the discovery of cocyclic Hadamard matrices by Warwick de Launey and myself. His effort to characterise those Hadamard matrices which would generate higher dimensional proper Hadamard matrices led him to isolate functions which must satisfy specific relations between their values and which I subsequently identified as cocycles.

A 2-dimensional cocycle between finite groups G and N, with trivial action, is a function [psi] : G x G [right arrow] N satisfying the equation


We then rederived this equation by asking when an abstract combinatorial design could be functionally generated from a single row. This cocyclic development of matrices includes group development of matrices, which was described in Chapter 2. The cocyclic matrix developed from [psi] : G x G [right arrow] N is


A cocycle whose matrix is Hadamard is called orthogonal.

The first Chapter of Part 2, Chapter 6, concerns cocycles, which arise naturally in many areas: surface topology, algebra and quantum mechanics, for instance. The usual unit studied in group cohomology is a cohomology (equivalence) class of cocycles, not the individual cocycles comprising it, so the examples, properties and constructions collected here do not appear in cohomology texts and are listed for the first time.

Some time is spent on the practicalities of computing cocycles. One of the advantages of the group extensions approach to Hadamard matrices is that the internal structure of a cocyclic matrix promises efficiency in computer searches for generalised Hadamard matrices, cutting down the search space over exhaustion dramatically. But first we need to find and list the cocycles. Three algorithms are presented: one, the Flannery-O'Brien algorithm, was developed to exploit the ideas presented in this book and is distributed as a module in the computer algebra package MAGMA.

The Chapter continues by showing that most of the direct constructions of Hadamard matrices listed in Chapter 2 are cocyclic, for some group G and N = {±1}. To date, cocyclic construction is the most successful general method known, both theoretical and computational, for finding Hadamard matrices. In particular, the most productive single construction of Hadamard matrices, due to Ito, is cocyclic over the dihedral groups. The Cocyclic Hadamard Conjecture follows: that for each odd t there is a group G of order 4t such that a G- cocyclic Hadamard matrix exists. The Chapter concludes with a status report on 12 research questions posed by the author in earlier papers on cocyclic Hadamard matrices.

Cocycles are special cases of factor pairs of functions. Chapter 7 contains the full description of the theory of orthogonal factor pairs and the generalised Hadamard matrices they determine. The theory has been complete for only a few years. Sufficient background information on group extensions, factor pairs and cohomology of finite groups is included to make the book self-contained.

The limiting class of generalised Hadamard matrices obtained using the group extensions approach is the class of coupled cocyclic generalised Hadamard matrices. We can do no better than this. Whilst not every generalised Hadamard matrix is a coupled cocyclic matrix, I know of only one counterexample, a matrix of order 6 with entries from the group [[??].sub.3] of integers modulo 3. I know of no Hadamard matrix which is not cocyclic - but the sheer number of inequivalent Hadamard matrices even for small orders makes it unlikely all will be cocyclic.

The Chapter's central purpose is to convey the pervasive influence of cocyclic generalised Hadamard matrices, by locating them (in four different guises) within combinatorics, group cohomology, incidence structures and digital sequence design. This is done by proving mutual equivalences - the Five-fold Constellation - between coupled cocyclic generalised Hadamard matrices, semiregular relative difference sets, orthogonal factor pairs, semiregular class regular divisible designs with regular action and well-correlated arrays. These equivalences have been established in increasing generality over the past decade by de Launey, Flannery, Perera, Hughes and the author, with the fullest expression due to Galati. The general form of the fifth equivalence - with well-correlated arrays - is given here for the first time. Such universality helps to explain the tremendous variety of uses to which we can put these matrices.

Chapter 8 deals with the way in which different definitions of equivalence class interrelate within the Five-fold Constellation. There are preexisting concepts of equivalence for generalised Hadamard matrices, for transversals of subgroups in groups, and for factor pairs and group extensions arising naturally from theoretical considerations in each area, and they do not coincide. The equivalence relation for transversals is revealed to be the strongest relation. It becomes a very productive and novel way of investigating each of the 'stars' of the Constellation.

When equivalence of transversals is transcribed to an action on factor pairs, it forms orbits termed bundles. These bundles are copied around the Five-fold Constellation. For splitting factor pairs, bundles define equivalence classes of functions G [right arrow] N, which form the basis of a new theory of nonlinearity. For semiregular relative difference sets, the resulting taxonomy allows us to establish a classification program for their equivalence classes and begin to populate it. This problem is at the heart of research in relative difference sets.

Two components of bundle action can be isolated, one an action by automorphism groups of G and N and the other a differential G-action called shift action which arises from translation and renormalisation of transversals. Thus a bundle is an automorphism orbit of shift orbits, and vice versa. These components, though not wholly independent, can be extracted and investigated in more general situations.

Shift action is a remarkably universal action and should be identifiable in more contexts than in fact appears to be the case. Shift action operates wholly within the natural equivalence classes of factor pairs, partitioning each one into shift orbits - its atomic structure. So, it is invisible from the point of view of cohomology theory, but it is critical to our study. Shift orbits (and the bundles they generate) carry the statistical information about distributions of the entries of cocyclic matrices that determines whether or not they will produce Hadamard matrices, high-distance error-correcting codes and low-correlation sequences.

Some external sightings of shift action in disguise have been made: in differential cryptanalysis and in the Loewy series for p-groups. LeBel's thesis [217] identifies shift action within the trivial cohomology class with a natural action in a quotient algebra of the standard module of a group ring.

In the final, and longest, Chapter, we begin to reap the rewards of all the preceding hard work. Chapter 9 contains a multitude of new constructions and applications of cocyclic complex and generalised Hadamard matrices, and a tantalising set of new problems, too.

Initially we look at several recent applications of cocycles, not necessarily orthogonal, to computation in Galois rings, to elliptic curve cryptography and to the developing field of cocyclic codes over nonbinary alphabets.

Then splitting orthogonal factor pairs are applied to establish a general theory of nonlinear functions suitable for use as cryptographic primitives. These include planar, bent and maximally nonlinear functions, and surprising and beautiful connections with finite presemifields and projective planes are uncovered. (Continues...)

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Table of Contents

Preface xi

Chapter 1. Introduction 1


Chapter 2. Hadamard Matrices 9
2.1 Classical Constructions 10
2.1.1 Sylvester Hadamard matrices 11
2.1.2 Paley Hadamard matrices 11
2.1.3 Hadamard designs 12
2.1.4Williamson Hadamard matrices 15
2.2 Equivalence Classes 16
2.3 The First Link: Group Developed Constructions 20
2.3.1 Menon Hadamard matrices 21
2.3.2 Ito Hadamard matrices 23
2.4 Towards the Hadamard Conjecture 25

Chapter 3. Applications in Signal Processing, Coding and Cryptography 27
3.1 Spectroscopy: Walsh-Hadamard Transforms 28
3.1.1 Signal analysis and synthesis 28
3.1.2 The Walsh-Hadamard Transform 29
3.1.3 The Fast Hadamard Transform 33
3.1.4 Hadamard spectroscopy 33
3.2 Error Correction: Hadamard Codes 35
3.2.1 Error-correcting codes 36
3.2.2 Hadamard codes 39
3.3 Signal Modulation and Separation: Hadamard Codes 43
3.3.1 CDMA for mobile, wireless and optical communications 45
3.3.2 3-D holographic memory for data storage and retrieval 47
3.4 Signal Correlation: Perfect Sequences and Arrays 48
3.4.1 Timing and synchronisation: Perfect binary sequences 49
3.4.2 Signal array correlation: Perfect binary arrays 50
3.5 Cryptography: Nonlinear Functions 53
3.5.1 Binary bent functions and maximally nonlinear functions 55
3.5.2 Perfect and almost perfect nonlinear functions 59

Chapter 4. Generalised Hadamard Matrices 62
4.1 Butson Matrices 63
4.2 Complex Hadamard Matrices 66
4.2.1 Quaternary complex Hadamard matrices 67
4.2.2 Unimodular complex Hadamard matrices 69
4.3 Generalised Hadamard Matrices 70
4.3.1 Generalised Hadamard matrix constructions 71
4.3.2 Generalised Hadamard matrices and Butson matrices 73
4.3.3 Generalised Hadamard matrices and class regular divisible designs 74
4.3.4 Group developed GH(w; v=w) and semiregular relative difference sets 75
4.4 Applications of Complex and Generalised Hadamard Matrices 78
4.4.1 Quaternary complex Hadamard transforms 78
4.4.2 Perfect quaternary sequences and arrays 79
4.4.3 Quaternary error-correcting codes 81
4.4.4 Generalised Hadamard matrices and Hadamard codes 83
4.5 Unification: Generalised Butson Hadamard Matrices and Transforms 84
4.5.1 The jacket matrix construction 85
4.5.2 The Generalised Hadamard Transform 90

Chapter 5. Higher Dimensional Hadamard Matrices 92
5.1 Classical Constructions 94
5.1.1 Boolean function construction for order 2 95
5.1.2 Product construction 97
5.1.3 Group developed construction 97
5.1.4 Perfect binary array construction 98
5.2 Equivalence Classes 99
5.3 Applications in Spectroscopy, Coding and Cryptography 100
5.3.1 Multidimensional Walsh Hadamard transforms 101
5.3.2 Error-correcting array codes 102
5.3.3 Cryptography: bent functions and the strict avalanche criterion 105
5.4 The Second Link: Cocyclic Construction 106


Chapter 6. Cocycles and Cocyclic Hadamard Matrices 113
6.1 Cocycles and Group Cohomology 114
6.2 Cocycles are Everywhere! 116
6.2.1 Examples of cocycles 116
6.2.2 New from old 117
6.2.3 Characteristic properties 119
6.2.4 Orthogonality and its inheritance 121
6.3 Computation of Cocycles 122
6.3.1 Algorithm 1— abelian groups 124
6.3.2 Algorithm 2— MAGMA implementation 126
6.3.3 Algorithm 3— Homological perturbation 127
6.4 Cocyclic Hadamard Matrices 128
6.4.1 Sylvester Hadamard matrices 128
6.4.2 Menon Hadamard matrices 129
6.4.3Williamson Hadamard matrices 129
6.4.4 Ito Hadamard matrices 129
6.4.5 Generalisations of Ito Hadamard matrices 130
6.4.6 Numerical results 131
6.5 The Cocyclic Hadamard Conjecture 133
6.5.1 Noncocyclic Hadamard matrix constructions? 134
6.5.2 Status report—research problems in cocyclic Hadamard matrices 137

Chapter 7. The Five-fold Constellation 139
7.1 Factor Pairs and Extensions 139
7.2 Orthogonality for Factor Pairs 143
7.3 All the Cocyclic Generalised Hadamard Matrices 146
7.3.1 Cocyclic generalised Hadamard matrix constructions 149
7.4 The Five-fold Constellation 151
7.4.1 Restrictions on existence of cocyclic generalised Hadamard matrices 158
7.4.2 Two approaches 160

Chapter 8. Bundles and Shift Action 162
8.1 Bundles and the Five-fold Constellation 163
8.1.1 Equivalence of transversals 163
8.1.2 Bundles of factor pairs 165
8.2 Bundles of Functions—The Splitting Case 170
8.3 Bundles of Cocycles—The Central Case 174
8.3.1 Automorphism action versus shift action 174
8.3.2 A taxonomy for central semiregular RDSs 176
8.3.3 Bundles with trivial shift action—the multiplicative cocycles 178
8.4 Shift Action—The Central Case 181
8.4.1 Orbit structure for cyclic groups 184
8.4.2 Relationship between orbit structures in distinct cohomology classes 185
8.5 Shift Orbits—The Central Splitting Case 185
8.5.1 When C is an elementary abelian p-group 187
8.5.2 When C is an elementary abelian p-group and G is a p-group 188

Chapter 9. The Future: Novel Constructions and Applications 192
9.1 New Applications of Cocycles 192
9.1.1 Computation in Galois rings 192
9.1.2 Elliptic curve cryptosystems 195
9.1.3 Cocyclic codes 197
9.1.4 Cocyclic Butson matrices and codes 202
9.2 New Group Developed Generalised Hadamard Matrices 204
9.2.1 Group developed GH matrices and PN functions 204
9.2.2 PN functions and a theory of highly nonlinear functions 208
9.3 New Cocyclic Generalised Hadamard Matrices 212
9.3.1 Direct sum constructions 212
9.3.2 Multiplicative orthogonal cocycles and presemifields 216
9.3.3 Swing action 224
9.4 New Hadamard Codes 225
9.4.1 Class A cocyclic Hadamard codes 225
9.4.2 Class B cocyclic Hadamard codes 227
9.4.3 Class C cocyclic Hadamard codes 229
9.5 New Highly Nonlinear Functions 230
9.5.1 1-D differential uniformity 230
9.5.2 Differential 2-row uniformity and APN functions 233
9.5.3 2-D total differential uniformity 235

Bibliography 238
Index 259

What People are Saying About This

Christos Koukouvinos

This book will surely be a welcome contribution to the existing literature. The book can be used by researchers in the early stages of their work and as a source for a course in the subject.
Christos Koukouvinos, National Technical University of Athens

Charles Colbourn

The material on applications of Hadamard matrices is, in a word, excellent. Researchers and students in communications, cryptography, and experimental design will find this to be an excellent source, while research mathematicians will find the second part just as interesting. The book is insightful and rewarding. It is definitely a book that I will have on my most accessible shelf.
Charles Colbourn, Arizona State University

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