Semi-regular relative difference sets with large forbidden subgroups

Abstract

Relative difference set Semi-regular relative difference set 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 2 × Z 2 3 . 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.

On abelian (2n,n,2n,2)-difference sets

“…This is also a generalization of one of T. Feng and Q. Xiang's (2008) [2] results in the abelian case.…”

“…(See [4].) Let K = G F (2 m+1 ) and K = K /G F (2) and let G = K × K be a group with multiplication defined by (a, b)(c, d) = (a + c, b + d + ac). Then D := K × {0} is a (2 m+1 , 2 m , 2 m+1 , 2)-difference set in G relative to U := {0} × K .…”

Relative (pn,p,pn,n)-difference sets with GCD(p,n)=1

“…As far as we know, there are only three constructions ( [3,4,8]) of semi-regular RDSs in groups of sizes not a prime power when the forbidden subgroup has size larger than 2. The RDSs constructed in [3,8] have parameters…”