Orthogonal Designs of Kharaghani Type: I

Orthogonal Designs of Kharaghani Type: II

H. Kharaghani, in "Arrays for orthogonal designs", J. Combin. Designs, 8 (2000), 166-173, showed how to use amicable sets of matrices to construct orthogonal designs in orders divisible by eight. We show how amicable orthogonal designs can be used to make amicable sets and so obtain infinite families of orthogonal designs in six variables in orders divisible by eight.

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.

The asymptotic existence of orthogonal designs

Given any -tuple ( s1, s2, . . . , s ) of positive integers, there is an integer N = N ( s1, s2, . . . , s ) such that an orthogonal design of order 2 ( s1 + s2 + · · ·+ s ) and type ( 2s1, 2 s2, . . . , 2 s ) exists, for each n ≥ N . This complements a result of Eades et al. which in turn implies that if the positive integers s1, s2, . . . , s are all highly divisible by 2, then there is a ful...

Orthogonal designs and MDS self-dual codes

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.