Semi-regular Relative Difference Sets with Large Forbidden Subgroups

2008/01/22 by Tao Feng, Feng, Tao, Qing Xiang +1
Mathematics · #05B10 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05B10

paper · pdf · doi:10.48550/arxiv.0801.3394

arxiv created 2008/01/22 · arxiv updated 2009/12/01

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 2p², and an abelian (4p,p,4p,4) relative difference set can only exist in the group ℤ₂²× ℤ₃². 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 (mod 4). When q=p is a prime, p>9, and p≡ 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.

Related

• Paley type partial difference sets in abelian groups
• Almost Difference Sets in Nonabelian Groups
• Non-existence of two types of partial difference sets
• Difference sets disjoint from a subgroup II: groups of order 4p²
• Nonabelian partial difference sets constructed using abelian techniques