New D-optimal designs via cyclotomy and generalised cyclotomy

D-optimal designs are n x n ±l-matrices where n == 2 mod 4 with maximum determinant. D-optimal designs obtained via circulant matrices are equivalent to 2-{ v; kl i k2 i k1 + k2 ~(v 1)} supplementary difference sets, where v = ~. We use cyclotomy to construct D-optimal designs, where v is a prime. We give a generalisation of cyclotomy and extend the cyclotomic techniques which enables use to find new D-optimal designs for composite numbers. In particular, we found, via computer-search, D-optimal designs for v = ~ = 7,13,19,21,31,33,37,41,43,61,73,85,91,93,113. The case v 85 = 5 x 17 is completely new. That is, D-optimal designs of order n 2v = 2 x 85 are given here for the first time.