Williamson Matrices and a Conjecture of Ito's

We point out an interesting connection between Williamson matrices and relative difference sets in nonabelian groups. As a consequence, we are able to show that there are relative (4t, 2, 4t, 2t)-difference sets in the dicyclic groups Q_8t = a, b|a^4t = b⁴ = 1, a^2t = b², b^-1ab = a^-1 for all t of the form t = 2^a · 10^b · 26^c · m with a, b, c 0, m 1 (mod 2), whenever 2m-1 or 4m-1 is a prime power or there is a Williamson matrix over _m. This gives further support to an important conjecture of Ito IT5 which asserts that there are relative (4t, 2, 4t, 2t)-difference sets in Q_8t for every positive integer t. We also give simpler alternative constructions for relative (4t, 2, 4t, 2t)-difference sets in Q_8t for all t such that 2t - 1 or 4t - 1 is a prime power. Relative difference sets in Q_8t with these parameters had previously been obtained by Ito IT1. Finally, we verify Ito's conjecture for all t 46.