Some new orthogonal designs in orders 32 and 40

Some new orthogonal designs in orders 32 and 40

A result of Robinson states that no OD(n; 1, 1, 1, 1, 1, n− 5) exists for n > 40. We complement this result by showing the existence of OD(n; 1, 1, 1, 1, 1, n− 5) for n = 32, 40. This includes a resolution to an old open problem regarding orthogonal designs of order 32 as well. We also obtain a number of new orthogonal designs of order 32.

Some new constructions of orthogonal designs

In this paper we construct OD(4pq(q+1); pq, pq, pq, pq, pq, pq, pq, pq) for each core order q ≡ 3(mod 4), r ≥ 0 or q = 1, p odd, p ≤ 21 and p ∈ {25, 49}, and COD(2q(q + 1); q, q, q, q) for any prime power q ≡ 1(mod 4) (including q = 1), r ≥ 0.

Orthogonal designs of order 32 and 64 via computational algebra

Some constructions for amicable orthogonal designs

Hadamard matrices, orthogonal designs and amicable orthogonal designs have a number of applications in coding theory, cryptography, wireless network communication and so on. Product designs were introduced by Robinson in order to construct orthogonal designs especially full orthogonal designs (no zero entries) with maximum number of variables for some orders. He constructed product designs of o...

All triples for orthogonal designs of order 40

The use of amicable sets of eight circulant matrices and four negacyclic matrices to generate orthogonal designs is relatively new. We find all 1841 possible orthogonal designs of order 40 in three variables, using only these new techniques.