Cocyclic Generalised Hadamard Matrices and Central Relative Difference Sets

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Abstract

Cocyclic matrices have the form M = [ψ(g, h)]g, h ∈ G, where G is a finite group, C is a finite abelian group and ψ: G × G → C is a (two-dimensional) cocycle; that is, ψ(g, h) ψ(gh, k) = ψ(g, hk) ψ(h, k), ∀ g, h, k ∈ 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 subgroup C 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.

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