Constructing Hadamard matrices from orthogonal designs

The Hadamard conjecture is that Hadamard matrices exist for all orders 1,2, 4t where t 2 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 q for every s 2 [2log2 (q 3)]. (ii) Given any natural number q, there exists a regular symmetric Hadamard matrix with constant diagonal of order 226 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 2q for some c < co." We make steps toward proving the Hadamard conjecture by showing that "If there is an OD( 4pi S1, S2, S3, S4) and a set ofT-matrices of order t there is an OD(16pt; 4pts1 , 4pts2 , 4pts3, 4pts4). In particular, if there is an OD( 4pj p, p, p, p) and a set ofT-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 E {orders of Hadamard matrices, orders of conference matrices, 1 + 2"'10b26c , 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 E {73, 79, 83, 89, 97} plus other orders, and w for a number of infinite families. New Tsequences 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.