Cocyclic Generalised Hadamard Matrices and Central RelativeDifference Sets

Abstract

Cocyclic matrices have the form M = [ \psi(g, h)]_{g, h \in G}, where G is a finite group, C is a finite abelian group and \psi : G \times G \rightarrow C is a (two-dimensional) cocycle; that is, \psi(g, h) \psi(gh, k) = \psi(g, hk) \psi(h, k), \forall g, h, k \in G. This expression of the cocycle equation for finite groups as a square matrix allows us to link group cohomology, divisible designs with regular automorphism groups and relative difference sets. Let G have order v and C have order w, with w|v. We show that the existence of a G-cocyclic generalised Hadamard matrix GH (w, v/w) with entries in C is equivalent to the existence of a relative ( v, w, v, v/w)-difference set in a central extension E of C by G relative to the central subgroupC and, consequently, is equivalent to the existence of a (square) divisible ( v, w, v, v/w)-design, class regular with respect to C, with a central extension E of C as regular group of automorphisms. This provides a new technique for the construction of semiregular relative difference sets and transversal designs, and generalises several known results.