Cocyclic Generalised Hadamard Matrices and Central RelativeDifference Sets

• Authors: A. A. I. Perera;K. J. Horadam
• Venue: Designs, Codes and Cryptography
• Year: 1998

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