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Hadamard matrices and their applications. (English) Zbl 1145.05014
Princeton, NJ: Princeton University Press (ISBN 0-691-11921-X/hbk). xiii, 263 p. (2007).
This new book is organized into two parts, the first dealing with Hadamard matrices and their applications and the second part dealing with cocyclic Hadamard matrices. The familiar material on designs and classical constructions is dealt with in 25 pages (Chapter 2). The rest of the 263 pages deals with the modern theory, much of which has been developed by the author and her colleagues. Chapter 3 deals with applications in signal processing, coding and cryptography, the Walsh-Hadamard transform, the fast Hadamard transform, Hadamard spectroscopy, signal modulation and Hadamard codes. Chapters 4 and 5, on generalized and higher dimensional Hadamard matrices, complete Part 1.
A major purpose of these chapters is to prepare the reader for Part 2, Cocyclic Hadamard Matrices. Here, in the remaining four chapters, the reader is brought up to date on the most modern research applying group cohomology to signal processing, and the author takes great pains to point towards the future of each topic. There are 90 open problems listed by number. Many proofs are only sketched, usually with suitable citations to the literature (330 items are in the bibliography), and this makes the text very readable. It is highly recommended for graduate students and researchers in mathematics, computer science, and communications engineering, and reference libraries. For most of the book very little background is needed other than basic linear algebra and the rudiments of modern algebra. Elementary cohomology could be developed as needed in a graduate seminar for instance.
This book will be a standard reference for all those interested in these areas.

05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
94B25 Combinatorial codes
94A12 Signal theory (characterization, reconstruction, filtering, etc.)