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 2sq for every s ≥ [2log2(q - 3].
(ii) Given any natural number q, there exists a regular symmetric Hadamard matrix with constant diagonal of order 22s q2 for s as before.
A significant step towards proving the Hadamard conjecture would be proving "Given any natural number q and constant Co there exists a Hadamard matrix of order 2cq for some c < co."
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 + 2alOb26c, 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.