On orthogonal matrices

iv, 64 leaves : ill., map ; 29 cm.Our main aim in this thesis is to study and search for orthogonal matrices which have a certain kind of block structure. The most desirable class of matrices for our purpose are orthogonal designs constructible from 16 circulant matrices. In studying these matrices, we show that the OD (12;1,1,1,9) is the only orthogonal design constructible from 16 circulant matrices of type OD (4n;1,1,1,4n-3), whenever n > 1 is an odd integer. We then use an exhaustive search to show that the only orthogonal design constructible from 16 circulant matrices of order 12 on 4 variables is the OD (12;1,1,1,9). It is known that by using of T-matrices and orthogonal designs constructible from 16 circulant matrices one can produce an infinite family of orthogonal designs. To complement our studies we reproduce and important recent construction of T-matrices by Xia and Xia. We then turn our attention to the applications of orthogonal matices. In some recent works productive regular Hadamard matrices are used to construct many new infinite families of symmetric designs. We show that for each integer n for which 4n is the order of a Hadamard matrix and 8n2 - 1 is a prime, there is a productive regular Hadamard matrix of order 16n2(82-1)2. As a corollary, we get many new infinite classes of symmetric designs whenever either of 4n(8n2-1)-1,4n(82-1) +1 is a prime power. We also review some other constructions of productive regular Hadamard matrices which are related to our work

Classification of generalized Hadamard matrices H(6,3) and quaternary Hermitian self-dual codes of length 18

All generalized Hadamard matrices of order 18 over a group of order 3, H(6,3), are enumerated in two different ways: once, as class regular symmetric (6,3)-nets, or symmetric transversal designs on 54 points and 54 blocks with a group of order 3 acting semi-regularly on points and blocks, and secondly, as collections of full weight vectors in quaternary Hermitian self-dual codes of length 18. The second enumeration is based on the classification of Hermitian self-dual [18,9] codes over GF(4), completed in this paper. It is shown that up to monomial equivalence, there are 85 generalized Hadamard matrices H(6,3), and 245 inequivalent Hermitian self-dual codes of length 18 over GF(4).Comment: 17 pages. Minor revisio

Exotic complex Hadamard matrices, and their equivalence

In this paper we use a design theoretical approach to construct new, previously unknown complex Hadamard matrices. Our methods generalize and extend the earlier results of de la Harpe--Jones and Munemasa--Watatani and offer a theoretical explanation for the existence of some sporadic examples of complex Hadamard matrices in the existing literature. As it is increasingly difficult to distinguish inequivalent matrices from each other, we propose a new invariant, the fingerprint of complex Hadamard matrices. As a side result, we refute a conjecture of Koukouvinos et al. on (n-8)x(n-8) minors of real Hadamard matrices.Comment: 10 pages. To appear in Cryptography and Communications: Discrete Structures, Boolean Functions and Sequence