Characteristic Functions of Relative Difference Sets, Correlated Sequences and Hadamard Matrices

Abstract

Given a cocycle, {α}, the concept of a sequence being α-correlated provides a link between the cohomology of finite groups and various combinatorial objects: auto-correlated sequences, relative difference sets and generalised Hadamard matrices. The cohomology enables us to lift a map Φ, defined on a group, to a map Φ, defined on an extension group, in such a way that Φ inherits some of its combinatorial properties from those of Φ. For example, if Φ is α-correlated, Φ will be the characteristic function of a relative difference set in the extension group determined by α. Many well-known results follow from choosing the appropriate extension groups and cocycles, α.