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Hadamard Matrices and Their Applications - K. J. Horadam

Hadamard Matrices and Their Applications


Published: 13th November 2006
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In "Hadamard Matrices and Their Applications," K. J. Horadam provides the first unified account of cocyclic Hadamard matrices and their applications in signal and data processing. This original work is based on the development of an algebraic link between Hadamard matrices and the cohomology of finite groups that was discovered fifteen years ago. The book translates physical applications into terms a pure mathematician will appreciate, and theoretical structures into ones an applied mathematician, computer scientist, or communications engineer can adapt and use.

The first half of the book explains the state of our knowledge of Hadamard matrices and two important generalizations: matrices with group entries and multidimensional Hadamard arrays. It focuses on their applications in engineering and computer science, as signal transforms, spreading sequences, error-correcting codes, and cryptographic primitives.

The book's second half presents the new results in cocyclic Hadamard matrices and their applications. Full expression of this theory has been realized only recently, in the Five-fold Constellation. This identifies cocyclic generalized Hadamard matrices with particular "stars" in four other areas of mathematics and engineering: group cohomology, incidence structures, combinatorics, and signal correlation.

Pointing the way to possible new developments in a field ripe for further research, this book formulates and discusses ninety open questions.

"This book is a marvelous and timely contribution to a rapidly developing field, with new oflshoots into physics, engineering and algebra... Overall, the text gives an excellent introduction to Hadamard matrices, a masterful short survey of applications the field of communications, and a wild ride through these new algebraic tools and new combinatorial objects of study being spawned by this modern approach. Absolutely up-to-date and useful, his is a must-have text for all researchers in this field, and a must-read for aspiring researchers of Hadamard matrices, their generalizations, and their applications."--Robert Craigen, Mathematical Reviews

Prefacep. xi
Introductionp. 1
Hadamard Matrices, Their Applications and Generalisationsp. 7
Hadamard Matricesp. 9
Classical Constructionsp. 10
Sylvester Hadamard matricesp. 11
Paley Hadamard matricesp. 11
Hadamard designsp. 12
Williamson Hadamard matricesp. 15
Equivalence Classesp. 16
The First Link: Group Developed Constructionsp. 20
Menon Hadamard matricesp. 21
Ito Hadamard matricesp. 23
Towards the Hadamard Conjecturep. 25
Applications in Signal Processing, Coding and Cryptographyp. 27
Spectroscopy: Walsh-Hadamard Transformsp. 28
Signal analysis and synthesisp. 28
The Walsh-Hadamard Transformp. 29
The Fast Hadamard Transformp. 33
Hadamard spectroscopyp. 33
Error Correction: Hadamard Codesp. 35
Error-correcting codesp. 36
Hadamard codesp. 39
Signal Modulation and Separation: Hadamard Codesp. 43
CDMA for mobile, wireless and optical communicationsp. 45
3-D holographic memory for data storage and retrievalp. 47
Signal Correlation: Perfect Sequences and Arraysp. 48
Timing and synchronisation: Perfect binary sequencesp. 49
Signal array correlation: Perfect binary arraysp. 50
Cryptography: Nonlinear Functionsp. 53
Binary bent functions and maximally nonlinear functionsp. 55
Perfect and almost perfect nonlinear functionsp. 59
Generalised Hadamard Matricesp. 62
Butson Matricesp. 63
Complex Hadamard Matricesp. 66
Quaternary complex Hadamard matricesp. 67
Unimodular complex Hadamard matricesp. 69
Generalised Hadamard Matricesp. 70
Generalised Hadamard matrix constructionsp. 71
Generalised Hadamard matrices and Butson matricesp. 73
Generalised Hadamard matrices and class regular divisible designsp. 74
Group developed GH(w, v/w) and semiregular relative difference setsp. 75
Applications of Complex and Generalised Hadamard Matricesp. 78
Quaternary complex Hadamard transformsp. 78
Perfect quaternary sequences and arraysp. 79
Quaternary error-correcting codesp. 81
Generalised Hadamard matrices and Hadamard codesp. 83
Unification: Generalised Butson Hadamard Matrices and Transformsp. 84
The jacket matrix constructionp. 85
The Generalised Hadamard Transformp. 90
Higher Dimensional Hadamard Matricesp. 92
Classical Constructionsp. 94
Boolean function construction for order 2p. 95
Product constructionp. 97
Group developed constructionp. 97
Perfect binary array constructionp. 98
Equivalence Classesp. 99
Applications in Spectroscopy, Coding and Cryptographyp. 100
Multidimensional Walsh Hadamard transformsp. 101
Error-correcting array codesp. 102
Cryptography: bent functions and the strict avalanche criterionp. 105
The Second Link: Cocyclic Constructionp. 106
Cocyclic Hadamard Matricesp. 111
Cocycles and Cocyclic Hadamard Matricesp. 113
Cocycles and Group Cohomologyp. 114
Cocycles are Everywhere!p. 116
Examples of cocyclesp. 116
New from oldp. 117
Characteristic propertiesp. 119
Orthogonality and its inheritancep. 121
Computation of Cocyclesp. 122
Algorithm 1 - abelian groupsp. 124
Algorithm 2 - MAGMA implementationp. 126
Algorithm 3 - Homological perturbationp. 127
Cocyclic Hadamard Matricesp. 128
Sylvester Hadamard matricesp. 128
Menon Hadamard matricesp. 129
Williamson Hadamard matricesp. 129
Ito Hadamard matricesp. 129
Generalisations of Ito Hadamard matricesp. 130
Numerical resultsp. 131
The Cocyclic Hadamard Conjecturep. 133
Noncocyclic Hadamard matrix constructions?p. 134
Status report - research problems in cocyclic Hadamard matricesp. 137
The Five-fold Constellationp. 139
Factor Pairs and Extensionsp. 139
Orthogonality for Factor Pairsp. 143
All the Cocyclic Generalised Hadamard Matricesp. 146
Cocyclic generalised Hadamard matrix constructionsp. 149
The Five-fold Constellationp. 151
Restrictions on existence of cocyclic generalised Hadamard matricesp. 158
Two approachesp. 160
Bundles and Shift Actionp. 162
Bundles and the Five-fold Constellationp. 163
Equivalence of transversalsp. 163
Bundles of factor pairsp. 165
Bundles of Functions - The Splitting Casep. 170
Bundles of Cocycles - The Central Casep. 174
Automorphism action versus shift actionp. 174
A taxonomy for central semiregular RDSsp. 176
Bundles with trivial shift action - the multiplicative cocyclesp. 178
Shift Action - The Central Casep. 181
Orbit structure for cyclic groupsp. 184
Relationship between orbit structures in distinct cohomology classesp. 185
Shift Orbits - The Central Splitting Casep. 185
When C is an elementary abelian p-groupp. 187
When C is an elementary abelian p-group and G is a p-groupp. 188
The Future: Novel Constructions and Applicationsp. 192
New Applications of Cocyclesp. 192
Computation in Galois ringsp. 192
Elliptic curve cryptosystemsp. 195
Cocyclic codesp. 197
Cocyclic Butson matrices and codesp. 202
New Group Developed Generalised Hadamard Matricesp. 204
Group developed GH matrices and PN functionsp. 204
PN functions and a theory of highly nonlinear functionsp. 208
New Cocyclic Generalised Hadamard Matricesp. 212
Direct sum constructionsp. 212
Multiplicative orthogonal cocycles and presemifieldsp. 216
Swing actionp. 224
New Hadamard Codesp. 225
Class A cocyclic Hadamard codesp. 225
Class B cocyclic Hadamard codesp. 227
Class C cocyclic Hadamard codesp. 229
New Highly Nonlinear Functionsp. 230
1-D differential uniformityp. 230
Differential 2-row uniformity and APN functionsp. 233
2-D total differential uniformityp. 235
Bibliographyp. 238
Indexp. 259
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780691119212
ISBN-10: 069111921X
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 263
Published: 13th November 2006
Country of Publication: US
Dimensions (cm): 24.18 x 16.51  x 2.39
Weight (kg): 0.52