#### 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^{s}q for every s ≥ [2log_{2}(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^{c}q for some c < c_{o}."

We make steps toward proving the Hadamard conjecture by showing that "If there is an OD(4p; s_{1}, s_{2}, s_{3}, s_{4}) and a set of T-matrices of order t there is an OD(16p^{2}t; 4ptS_{1}, 4pts_{2}, 4pts_{3}, 4pts_{4}). 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(16p^{2}t; 4p^{2}t, 4p^{2}t, 4p^{2}t, 4p^{2}t). Further, if there are Williamson matrices of order w there is a Hadamard matrix of 16p^{2}tw."

Currently the aforementioned matrices are known for p, t є {orders of Hadamard matrices, orders of conference matrices, 1 + 2^{a}lO^{b}26^{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.

## Publication Details

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