On the nonexistence of semi-regular relative difference sets

Abstract

In this paper, we study semi-regular relative difference sets. We give some nonexistence results on abelian (mn,n,mn,m) relative difference sets. In particular, we focus on the case when m is prime and show that, for any fixed integer n≥2, there are at most finitely many primes p for which an abelian (pn,n,pn,p) relative difference set may exist. We illustrate our results by investigating the existence of (mn,n,mn,m) relative difference sets with m∈{2,3,4} in detail.