A New Construction of Central Relative ( p a , p a , p a , 1)-Difference Sets

Abstract

Semiregular relative difference sets (RDS) in a finite group E which avoid a central subgroup C are equivalent to orthogonal cocycles. For example, every abelian semiregular RDS must arise from a symmetric orthogonal cocycle, and vice versa. Here, we introduce a new construction for central (pa, pa, pa, 1)-RDS which derives from a novel type of orthogonal cocycle, an LP cocycle, defined in terms of a linearised permutation (LP) polynomial and multiplication in a finite presemifield. The construction yields many new non-abelian (pa, pa, pa, 1)-RDS. We show that the subset of the LP cocycles defined by the identity LP polynomial and multiplication in a commutative semifield determines the known abelian (pa, pa, pa, 1)-RDS, and give a second new construction using presemifields. We use this cohomological approach to identify equivalence classes of central (pa, pa, pa, 1)-RDS with elementary abelian C and E/C. We show that for p e 2, a ≤ 3 and p e 3, a ≤ 2, every central (pa, pa, pa, 1)-RDS is equivalent to one arising from an LP cocycle, and list them all by equivalence class. For p e 2, a e 4, we list the 32 distinct equivalence classes which arise from field multiplication. We prove that, for any p, there are at least a equivalence classes of central (pa, pa, pa, 1)-RDS, of which one is abelian and a − 1 are non-abelian.