Semi-regular relative difference sets with large forbidden subgroups

Authors:
Tao Feng;Qing Xiang
Affiliations:
School of Mathematical Sciences, Peking University, Beijing 100871, China;Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA
Venue:
Journal of Combinatorial Theory Series A
Year:
2008

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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 (2p,p,2p,2) relative difference set in any group of order 2p^2, and an abelian (4p,p,4p,4) relative difference set can only exist in the group Z"2^2xZ"3^2. On the other hand, we construct a family of non-abelian relative difference sets with parameters (4q,q,4q,4), where q is an odd prime power greater than 9 and q=1(mod4). When q=p is a prime, p9, and p=1(mod4), the (4p,p,4p,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.