# Williamson Matrices and a Conjecture of Ito's

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## Abstract

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 (4*t*, 2, 4*t*, 2t)-difference sets in the dicyclic groups *Q*_{8t} = 〈*a*, *b*|*a*^{4t} = *b*^{4} = 1, *a*^{2t} = *b*^{2}, *b*^{-1}ab = 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 2*m*-1 or 4*m*-1 is a prime power or there is a Williamson matrix over ℤ_{m}. This gives further support to an important conjecture of Ito IT5 which asserts that there are relative (4*t*, 2, 4*t*, 2*t*)-difference sets in *Q*_{8t} for every positive integer *t*. We also give simpler alternative constructions for relative (4*t*, 2, 4*t*, 2*t*)-difference sets in *Q*_{8t} for all *t* such that 2*t* - 1 or 4*t* - 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.

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