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Hadamard Matrices and Their Applications 1st Edition by KJ Horadam ISBN 1400842905 9781400842902

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Author(s): K. J. Horadam
ISBN(s): 9781400842902, 1400842905
Edition: Core Textbook
File Details: PDF, 1.22 MB
Year: 2012
Language: english
SKU: EB-51956488 Category: Tag:

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ISBN 10: 1400842905 
ISBN 13: 9781400842902
Author: KJ Horadam

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.

Hadamard Matrices and Their Applications 1st Table of contents:

Chapter 1. Introduction
PART 1. HADAMARD MATRICES, THEIR APPLICATIONS AND GENERALISATIONS
Chapter 2. Hadamard Matrices
2.1 Classical Constructions
2.2 Equivalence Classes
2.3 The First Link: Group Developed Constructions
2.4 Towards the Hadamard Conjecture
Chapter 3 Applications in Signal Processing, Coding and Cryptography
3.1 Spectroscopy: Walsh-Hadamard Transforms
3.2 Error Correction: Hadamard Codes
3.3 Signal Modulation and Separation: Hadamard Codes
3.4 Signal Correlation: Perfect Sequences and Arrays
3.5 Cryptography: Nonlinear Functions
Chapter 4 Generalised Hadamard Matrices
4.1 Butson Matrices
4.2 Complex Hadamard Matrices
4.3 Generalised Hadamard Matrices
4.4 Applications of Complex and Generalised Hadamard Matrices
4.5 Unification: Generalised Butson Hadamard Matrices and Transforms
Chapter 5 Higher Dimensional Hadamard Matrices
5.1 Classical Constructions
5.2 Equivalence Classes
5.3 Applications in Spectroscopy, Coding and Cryptography
5.4 The Second Link: Cocyclic Construction
PART 2: COCYCLIC HADAMARD MATRICES
Chapter 6 Cocycles and Cocyclic Hadamard Matrices
6.1 Cocycles and Group Cohomology
6.2 Cocycles are Everywhere!
6.3 Computation of Cocycles
6.4 Cocyclic Hadamard Matrices
6.5 The Cocyclic Hadamard Conjecture
Chapter 7 The Five-fold Constellation
7.1 Factor Pairs and Extensions
7.2 Orthogonality for Factor Pairs
7.3 All the Cocyclic Generalised Hadamard Matrices
7.4 The Five-fold Constellation
Chapter 8 Bundles and Shift Action
8.1 Bundles and the Five-fold Constellation
8.2 Bundles of Functions— The Splitting Case
8.3 Bundles of Cocycles— The Central Case
8.4 Shift Action — The Central Case
8.5 Shift Orbits — The Central Splitting Case
Chapter 9 The Future: Novel Constructions and Applications
9.1 New Applications of Cocycles
9.2 New Group Developed Generalised Hadamard Matrices
9.3 New Cocyclic Generalised Hadamard Matrices
9.4 New Hadamard Codes
9.5 New Highly Nonlinear Functions

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