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.

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.

Drawing on the authors’ use of the Hadamard-related theory in several successful engineering projects, Theory and Applications of Higher-Dimensional Hadamard Matrices, Second Edition explores the applications and dimensions of Hadamard matrices. This edition contains a new section on the applications of higher-dimensional Hadamard matrices to the areas of telecommunications and information security. The first part of the book presents fast algorithms, updated constructions, existence results, and generalized forms for Walsh and Hadamard matrices. The second section smoothly transitions from two-dimensional cases to three-, four-, and six-dimensional Walsh and Hadamard matrices and transforms. In the third part, the authors discuss how the n-dimensional Hadamard matrices of order 2 are applied to feed-forward networking, stream ciphers, bent functions, and error correcting codes. They also cover the Boolean approach of Hadamard matrices. The final part provides examples of applications of Hadamard-related ideas to the design and analysis of one-dimensional sequences and two-dimensional arrays. The theory and ideas of Hadamard matrices can be used in many areas of communications and information security. Through the research problems found in this book, readers can further explore the fascinating issues and applications of the theory of higher-dimensional Hadamard matrices.

Up-to-date resource on Hadamard matrices Hadamard Matrices: Constructions using Number Theory and Algebra provides students with a discussion of the basic definitions used for Hadamard Matrices as well as more advanced topics in the subject, including: Gauss sums, Jacobi sums and relative Gauss sums Cyclotomic numbers Plug-in matrices, arrays, sequences and M-structure Galois rings and Menon Hadamard differences sets Paley difference sets and Paley type partial difference sets Symmetric Hadamard matrices, skew Hadamard matrices and amicable Hadamard matrices A discussion of asymptotic existence of Hadamard matrices Maximal determinant matrices, embeddability of Hadamard matrices and growth problem for Hadamard matrices The book can be used as a textbook for graduate courses in combinatorics, or as a reference for researchers studying Hadamard matrices. Utilized in the fields of signal processing and design experiments, Hadamard matrices have been used for 150 years, and remain practical today. Hadamard Matrices combines a thorough discussion of the basic concepts underlying the subject matter with more advanced applications that will be of interest to experts in the area.

Foremost experts in their field have contributed articles resulting in a compilation of useful and timely surveys in this ever-expanding field. Each of these 12 original papers covers important aspects of design theory including several in areas that have not previously been surveyed. Also contains surveys updating earlier ones where research is particularly active.

This volume develops the depth and breadth of the mathematics underlying the construction and analysis of Hadamard matrices, and their use in the construction of combinatorial designs. At the same time, it pursues current research in their numerous applications in security and cryptography, quantum information, and communications. Bridges among diverse mathematical threads and extensive applications make this an invaluable source for understanding both the current state of the art and future directions.​ ​The existence of Hadamard matrices remains one of the most challenging open questions in combinatorics. Substantial progress on their existence has resulted from advances in algebraic design theory using deep connections with linear algebra, abstract algebra, finite geometry, number theory, and combinatorics. Hadamard matrices arise in a very diverse set of applications. Starting with applications in experimental design theory and the theory of error-correcting codes, they have found unexpected and important applications in cryptography, quantum information theory, communications, and networking.