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 Q8t = 〈a, b|a4t = b4 = 1, a2t = b2, b-1ab = a-1〉 for all t of the form t = 2a · 10b · 26c · 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 ℤm. This gives further support to an important conjecture of Ito IT5 which asserts that there are relative (4t, 2, 4t, 2t)-difference sets in Q8t for every positive integer t. We also give simpler alternative constructions for relative (4t, 2, 4t, 2t)-difference sets in Q8t for all t such that 2t - 1 or 4t - 1 is a prime power. Relative difference sets in Q8t with these parameters had previously been obtained by Ito IT1. Finally, we verify Ito's conjecture for all t ≤ 46.