Construction and classification of group invariant Butson Hadamard matrices

Abstract

The purpose of this dissertation is to introduce the reader to the study of group invariant Butson Hadamard matrices and to present our contribution in this area. Our focus is on abelian p-groups since one can obtain group invariant Butson Hadamard matrices for larger groups by using Kronecker products. The main areas of study we have identified are: • The existence and the construction of group invariant Butson Hadamard matrices • The non-existence and the necessary conditions for their existence • The classification of group invariant Butson Hadamard matrices Our contribution to this study is the outline of five types of constructions of group invariant Butson Hadamard matrices, a new necessary condition for existence under the self-conjugacy assumption and the classification of Butson Hadamard matrices which are invariant under the groups ℤₚ × ℤₚ and ℤₚ₂ × ℤₚ × ℤₚ.