Asymptotic existence of orthogonal designs

Abstract

An orthogonal design of order n and type (si,..., se), denoted OD(n; si,..., se), is a square matrix X of order n with entries from {0, ±x1,..., ±xe}, where the Xj’s are commuting variables, that satisfies XX* = ^ ^g=1 sjx^j In, where X* denotes the transpose of X, and In is the identity matrix of order n.

An asymptotic existence of orthogonal designs is shown. More precisely, for any Atuple (s1,..., se) of positive integers, there exists an integer N = N(s1,..., se) such that for each n > N, there is an OD(2n(s1 + ... + se); 2ns1,..., 2nse). This result of Chapter 5 complements a result of Peter Eades et al. which in turn implies that if the positive integers s1, s2,..., se are all highly divisible by 2, then there is a full orthogonal design of type (s1, s2,..., se).

Some new classes of orthogonal designs related to weighing matrices are obtained in Chapter 3.

In Chapter 4, we deal with product designs and amicable orthogonal designs, and a construction method is presented.

Signed group orthogonal designs, a natural extension of orthogonal designs, are introduced in Chapter 6. Furthermore, an asymptotic existence of signed group orthogonal designs is obtained and applied to show the asymptotic existence of orthogonal designs.