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- W632223205 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. Disciplines Physical Sciences and Mathematics Publication Details Seberry, J, On Hadamard matrices of order 2tpq:I, Ars Combinatoria, 23B, 1987, 195-213. This journal article is available at Research Online: http://ro.uow.edu.au/infopapers/1025 On Hadamard Matrices of Order 2' pq:f" @default.
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- W632223205 date "1987-01-01" @default.
- W632223205 modified "2023-09-26" @default.
- W632223205 title "On Hadamard matrices of order 2t pq:I" @default.
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