Cyclotomy, Hadamard arrays and supplementary difference sets

A 4n x 4n Hadamard array, H, is a square matrix of order 4n with elements ± A, ± B, ± C, ± D each repeated n times in each row and column. Assuming the indeterminates A, B, C, D commute, the row vectors of H must be orthogonal. These arrays have been found for n = 1 (Williamson, 1944), n = 3 (Baumert-Hall, 1965), n = 5 (Welch, 1971), and some other odd n < 43 (Cooper, Hunt, Wallis). The results for n = 25, 31, 37, 41 are presented here, as is a result for n = 9 not based on supplementary difference sets. This gives the following new orders for Hadamard matrices < 4000: 1804, 3404, 3596, 3772. These results were obtained by using an adaption of cyclotomy which allows the product of incidence matrices to be easily derived. This adaption is developed and the constructions shown for some families of supplementary difference sets.