On (pa, p, pa, pa-1)-relative difference sets

Abstract

Abelian relative difference sets of parameters (m, n, k, λ)=(pa, p, pa, pa-1)are studied in this paper. In particular, we show that for an abelian group G of order p2c+1and a subgroup N of G of order p, a (p2c, p, p2c, p2c-1)-relative difference set exists in G relative to N if and only if exp (G)≤pc+1.Furthermore, we have some structural results on (p2cp, p2c, p2c-1)-relative difference sets in abelian groups of exponent pc+1. We also show that for an abelian group G of order 22 c+2 and a subgroup N of G of order 2, a (22 c+1, 2, 22 c+1, 22 c)-relative difference set exists in G relative to N if and only if exp(G)≤2c+2 and N is contained in a cyclic subgroup of G of order 4. New constructions of (p2c+1, p, p2c+1, p2c)-relative difference sets, where p is an odd prime, are given. However, we cannot find the necessary and sufficient condition for this case. © 1995 Kluwer Academic Publishers.