# Hadamard Matrices and Their Applications

Pages: 278
https://www.jstor.org/stable/j.ctt7t6pw

1. Front Matter
(pp. i-vi)
(pp. vii-x)
3. Preface
(pp. xi-xiv)
4. Chapter One Introduction
(pp. 1-6)

The purpose of this book is three-fold: to report the current status of existence and construction problems for Hadamard matrices and their generalisations; to give an accessible account of the new unifying approach to these problems using group cohomology; and to support an understanding of how these ideas are applied in digital communications. I have tried to present results and open problems with sufficient rigour, and direction to the literature, to enable readers to begin their own research, but with enough perspective for them to gain an overview without needing in-depth knowledge of the algebraic background.

The book has two...

5. ### PART 1. HADAMARD MATRICES, THEIR APPLICATIONS AND GENERALISATIONS

(pp. 9-26)

AHadamard matrix of order nis ann×nmatrixHwith entries from {±1} such that$H{{H}^{\text{T}}}=n{{I}_{n}}, \caption {(2.1)}$that is, for which the real inner product of any pair of distinct rows is 0. Examples for the smallest ordersn= 1, 2 and 4 are$\left[ 1 \right],\quad \left[ \begin{array}{cr} 1 & 1 \\ 1 & -1 \\ \end{array} \right],\quad \left[ \begin{array}{crrr} 1 & 1 & 1 & 1 \\ 1 & -1 & 1 & -1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & -1 & 1 \\ \end{array} \right].$

Hadamard matrices have excited interest for almost 150 years, since the first examples were published by Sylvester in 1867 [303]. Sylvester also noted that ifHis a Hadamard matrix, so is$\left[ \begin{array}{cr} H &H \\ H& -H\end{array} \right]; \caption {(2.2)}$the examples above illustrate two iterations of this construction. Then, in 1893, Hadamard [133] published examples in orders...

• Chapter Three Applications in Signal Processing, Coding and Cryptography
(pp. 27-61)

Practical application of Hadamard matrices goes back to 1937, when Yates developed an algorithm (a fast Hadamard transform) to determine which factors contributed the main effects in a factorial experiment.

Since then, most direct use of Hadamard matrices has fallen into one of three broad categories: for design of experiments, including factorial designs; for Hadamard transform spectroscopy and object recognition; and for coding of digital signals.

In experimental design, Hadamard matrices are building blocks for 2-level orthogonal arrays of strengths 2 or 3 (the Hadamard arrays), given, respectively, by the binary matricesAnandCnof Definition 3.13 below. The...

• Chapter Four Generalised Hadamard Matrices
(pp. 62-91)

It takes no great stretch of the imagination to ask what happens when the entries of a Hadamard matrix are allowed to have more values than ±1. For instance, Hadamard’s original interest in matrices with entries from the unit disc makes an extension to complex entries on the unit circle wholly natural.

But what makes a matrix intrinsically ‘Hadamard’? Is it a property inherent in the invertibility of Hadamard matrices, as square matrices with entries from a field? Apart from its independent interest, this characteristic drives the applications of Hadamard matrices as digital signal transforms. Is it a property inherent...

• Chapter Five Higher Dimensional Hadamard Matrices
(pp. 92-110)

As long ago as 1971, Shlichta discovered the existence of higher dimensional (±1) arrays with a range of orthogonality properties [292]. In particular, he constructed 3-dimensional cubical arraysA= [a(i1,i2,i3)] with the property that any 2-D subarray, obtained by fixing an index in one dimension, is a Hadamard matrix.

By the time of Shlichta’s discovery, Hadamard matrices had already been implemented very successfully in a variety of practical applications. The concurrent development, early in the 1970s, of image processing techniques (especially for television), the publication by Harmuth [141] of methods of applying 2- and 3-D Walsh functions...

6. ### PART 2. COCYCLIC HADAMARD MATRICES

• Chapter Six Cocycles and Cocyclic Hadamard Matrices
(pp. 113-138)

Cocycles occur naturally in many areas. The underlying ideas have been investigated for well over a century. Möbius introduced the concept of 2-complexes into surface topology in the mid-1860s, and Mayer introduced the corresponding algebraic chain complexes, in whichn-cocycles mapn-simplexes into abelian groups, in the 1920s and 1930s [236, II.9]. During the latter period Schreier [282, 283], Baer [15], Hall and Fitting also considered 2-cocycles (known as ’factor sets’) in the study of group extensions, following their introduction by Schur [284, 285, 286] in projective representation theory at the beginning of the twentieth century [236, IV.11]. In the...

• Chapter Seven The Five-fold Constellation
(pp. 139-161)

The full theoretical framework rising in this Chapter is due to Galati [118, 120], building on work of the author, de Launey, Flannery, Perera and Hughes, and the treatment here follows his. However, the material in Section 7.1 is standard, and more details may be found in texts such as [3, 276]. In this Section, the class of cocycles is expanded to its limit, the class offactor pairs, within the theory of group extensions, and basic properties of factor pairs are noted. As usual, we are interested in equivalence classes of factor pairs, particularly equivalence classes of thesplitting...

• Chapter Eight Bundles and Shift Action
(pp. 162-191)

If orthogonality were an easily identified property of factor pairs, the quest for orthogonal factor pairs with which to construct coupled cocyclic generalised Hadamard matrices might be simple.

The correspondence between semiregular relative difference sets and orthogonal factor pairs (Theorem 7.29) is a special case of the correspondence between transversals and factor pairs (Lemmas 7.7, 7.8). The natural equivalence relation on transversals preserves semiregular relative difference sets, and it could be hoped that the natural equivalence relation on factor pairs (Definition 7.4) would preserve orthogonality.

However, even for cocycles, the natural equivalence relation (cohomology) does not preserve orthogonality. Over the...

• Chapter Nine The Future: Novel Constructions and Applications
(pp. 192-237)

This Chapter is a rich storehouse of examples, applications and problems. One third of the open research problems appear here.

Initially we look at several recent uses of cocycles, not necessarily orthogonal, for computation in Galois rings, for cryptography using elliptic curves and for coding over nonbinary alphabets.

In Section 9.2, splitting orthogonal factor pairs are applied to lay the foundations for a general theory of nonlinear functions. These include planar, bent and maximally nonlinear functions, and in Section 9.3, surprising and beautiful connections with finite presemifields are uncovered. A useful technique for forming new orthogonal cocycles as direct sums...

7. Bibliography
(pp. 238-258)
8. Index
(pp. 259-263)