Recent advances in the construction of Hadamard matrices

In the past few years exciting new discoveries have been made in constructing Hadamard matrices. These discoveries have been centred in two ideas: (i) the construction of Baumert-Hall arrays by utilizing a construction of L. R. Welch, and (ii) finding suitable matrices to put into these arrays. We discuss these results, many of which are due to Richard J. Turyn or the author

A survey of base sequences, disjoint complementary sequences and OD(4t; t, t, t, t)

We survey the existence of base sequences, that is four sequences of lengths m + p, m + p, m, m, p odd with zero auto correlation function which can be used with Yang numbers and four disjoint complementary sequences (and matrices) with zero non-periodic (periodic) autocorrelation function to form longer sequences. We survey their application to make orthogonal designs OD(4t; t, t, t, t). We give the method of construction of OD(4t; t, t, t, t) for t = 1,3,..., 41, 45,...,65, 67, 69, 75, 77, 81, 85, 87, 91, 93, 95, 99, 101, 105, 111, 115, 117, 119, 123, 125, 129, 133, 141,..., 147, 153, 155, 159, 161, 165, 169, 171, 175, 177, 183, 185, 189, 195, 201, 203, 205, 209

The combinatorics of binary arrays

This paper gives an account of the combinatorics of binary arrays, mainly concerning their randomness properties. In many cases the problem reduces to the investigation on difference sets.postprin

Amicable matrices and orthogonal designs

This thesis is mainly concerned with the orthogonal designs of Baumert-Hall array type, OD(4n;n,n,n,n) where n=2k, k is odd integer. For every odd prime power p^r, we construct an infinite class of amicable T-matrices of order n=p^r+1 in association with negacirculant weighing matrices W(n,n-1). In particular, for p^r≡1 (mod 4) we construct amicable T-matrices of order n≡2 (mod 4) and application of these matrices allows us to generate infinite class of orthogonal designs of type OD(4n;n,n,n,n) and OD(4n;n,n,n-2,n-2) where n=2k; k is odd integer. For a special class of T-matrices of order n where each of T_i is a weighing matrix of weight w_i;1 ≤i≤4 and Williamson-type matrices of order m, we establish a theorem which produces four circulant matrices in terms of four variables. These matrices are additive and can be used to generate a new class of orthogonal design of type OD(4mn;w_1s,w_2s,w_3s,w_4s ); where s=4m. In addition to this, we present some methods to find amicable matrices of odd order in terms of variables which have an interesting application to generate some new orthogonal designs as well as generalized orthogonal designs.University of Lethbridge, NSER

Amicable T-matrices and applications

iii, 49 leaves ; 29 cmOur main aim in this thesis is to produce new T-matrices from the set of existing T-matrices. In Theorem 4.3 a multiplication method is introduced to generate new T-matrices of order st, provided that there are some specially structured T-matrices of orders s and t. A class of properly amicable and double disjoint T-matrices are introduced. A number of properly amicable T-matrices are constructed which includes 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 18, 22. To keep the new matrices disjoint an extra condition is imposed on one set of T-matrices and named double disjoint T-matrices. It is shown that there are some T-matrices that are both double disjoint and properly amicable. Using these matrices an infinite family of new T-matrices are constructed. We then turn our attention to the application of T-matrices to construct orthogonal designs and complex Hadamard matrices. Using T-matrices some orthogonal designs constructed from 16 circulant matrices are constructed. It is known that having T-matrices of order t and orthogonal designs constructible from 16 circulant matrices lead to an infinite family of orthogonal designs. Using amicable T-matrices some complex Hadamard matrices are shown to exist

Applications of Hadamard matrices, Journal of Telecommunications and Information Technology, 2003, nr 2

We present a number of applications of Hadamard matrices to signal processing, optical multiplexing, error correction coding, and design and analysis of statistics