Hadamard Matrices and Their Applications

Zusatztext "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 Informationen zum Autor K. J. Horadam Klappentext 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. Zusammenfassung Offers an account of cocyclic Hadamard matrices and their applications in signal and data processing. This work 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. Inhaltsverzeichnis 7895308 ...

"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

Autorentext
K. J. Horadam

Klappentext
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.

Zusammenfassung
Offers an account of cocyclic Hadamard matrices and their applications in signal and data processing. This work 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.

Inhalt
Preface xi Chapter 1. Introduction 1 PART 1. HADAMARD MATRICES, THEIR APPLICATIONS AND GENERALISATIONS 7 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 Constr…

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