Semi-regular relative difference sets with large forbidden subgroups

Abstract

Motivated by a connection between semi-regular relative difference sets and mutually unbiased bases, we study relative difference sets with parameters ( m , n , m , m / n ) in groups of non-prime-power orders. Let p be an odd prime. We prove that there does not exist a ( 2 p , p , 2 p , 2 ) relative difference set in any group of order 2 p 2 , and an abelian ( 4 p , p , 4 p , 4 ) relative difference set can only exist in the group Z 2 2 × Z 3 2 . On the other hand, we construct a family of non-abelian relative difference sets with parameters ( 4 q , q , 4 q , 4 ) , where q is an odd prime power greater than 9 and q ≡ 1 ( mod 4 ) . When q = p is a prime, p > 9 , and p ≡ 1 ( mod 4 ) , the ( 4 p , p , 4 p , 4 ) non-abelian relative difference sets constructed here are genuinely non-abelian in the sense that there does not exist an abelian relative difference set with the same parameters.