#### Abstract

We prove a new result for orthogonal designs showing if all full orthogonal designs, OD (r; a, b, r - a - b), exist, where gcd(a, b, r - a - b) = 2t, then all full orthogonal designs, OD(s; c, d, s - c - d), exist, where gcd(c, d, s - c - d) = 2t+u, u ≥ 0. It is known that,for infinitely many numbers r = 2wp,such OD(r; a, b, r - a - b) exist. In particular we show OD(4; x, y, 4 - x - y), OD(24; x, y, 24 - x - y) (and thus OD(24; 2x, 2y, 24 - 2x - 2y)) OD(40; 2x, 2y, 40 - 2x - 2y), OD(4.28; 4x, 4y, 4.28 - 4x - 4y), OD(8.36; 8x, 8y, 8.36 - 8x - 8y) and OD(16.44; 16x, 16y, 16.44 - 16x - 16y) exist. These orthogonal designs are used to show that Hadamard matrices can be constructed, for any odd q, in all these cases of order 2Wpq, where w>wo. In all cases this bound wo<[21og2(q -3)] (or its more precise counterparts given in the paper) is an improvement on previously known results. Moreover it is established that if p = 3, 5, 7, 9, 11, 15, 21, 27 then for the first 600 (or less for higher p) odd numbers pq, more than 97% of Hadamard matrices exist of order 2ypq with y ≤ 4. For example if p = 15, 21, 27 and q is an odd number < 500 then an Hadamard matrix of order 6Oq, 120q, 84q, 168q, l08q or 216q is known.

## Publication Details

Seberry, J, On Hadamard matrices of order 2tpq:I, Ars Combinatoria, 23B, 1987, 195-213.