Constructing Hadamard matrices from orthogonal designs

Christos Koukouvinos
Jennifer Seberry, University of Wollongong

Article comments

Jennifer Seberry and Christos Koukouvinos, Constructing Hadamard matrices from orthogonal designs, Australasian Journal of Combinatorics, 6, (1992), 267-278.

Abstract

The Hadamard conjecture is that Hadamard matrices exist for all orders 1,2, 4t where t ≥ 1 is an integer. We have obtained the following results which strongly support the conjecture:

(i) Given any natural number q, there exists an Hadamard matrix of order 2^sq for every s ≥ [2log₂(q - 3].

(ii) Given any natural number q, there exists a regular symmetric Hadamard matrix with constant diagonal of order 2^2s q2 for s as before.

A significant step towards proving the Hadamard conjecture would be proving "Given any natural number q and constant C_o there exists a Hadamard matrix of order 2^cq for some c < c_o."

We make steps toward proving the Hadamard conjecture by showing that "If there is an OD(4p; s₁, s₂, s₃, s₄) and a set of T-matrices of order t there is an OD(16p²t; 4ptS₁, 4pts₂, 4pts₃, 4pts₄). In particular, if there is an OD(4p;p,p,p,p) and a set of T-matrices of order t there is an OD(16p²t; 4p²t, 4p²t, 4p²t, 4p²t). Further, if there are Williamson matrices of order w there is a Hadamard matrix of 16p²tw."

Currently the aforementioned matrices are known for p, t є {orders of Hadamard matrices, orders of conference matrices, 1 + 2^alO^b26^c, a, b, c non-negative integers, 1,3,...,71,75,77,81,85,87,91,93,95,99} or for all orders of t ≤ 100 except possibly t є {73, 79, 83, 89, 97} plus other orders, and w for a number of infinite families. New T sequences for lengths 35, 61, 71, 183 and 671 are given.

This paper gives 36 new orders <40,000 for which Hadamard matrices exist. The current paper lends support to the belief that c ≤ 5.