Semi-regular Relative Difference Sets with Large Forbidden Subgroups

TL;DR

This paper investigates the existence and construction of semi-regular relative difference sets with specific parameters in groups of non-prime-power order, revealing non-existence results and providing new non-abelian constructions.

Contribution

It proves non-existence of certain relative difference sets in groups of order 2p^2 and characterizes abelian cases, while constructing new non-abelian relative difference sets for parameters involving odd prime powers.

Findings

No $(2p,p,2p,2)$ relative difference set exists in groups of order $2p^2$.

An abelian $(4p,p,4p,4)$ relative difference set exists only in $Z_2^2 imes Z_3^2$.

Constructed non-abelian $(4q,q,4q,4)$ relative difference sets for certain prime powers $q$.

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 $\Bbb{Z}_2^2\times \Bbb{Z}_3^2$. 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 $q\equiv 1$ (mod 4). When $q=p$ is a prime, $p>9$, and $p\equiv$ 1 (mod 4), 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.