Classification of difference matrices and complex Hadamard matrices

This dissertation gives a survey of algebraic methods for constructing generalized Hadamard matrices and for showing that generalized Hadamard matrices do not exist with the given parameters. Using these methods the existence or non-existence of generalized Hadamard matrices defined over an elementary abelian group is presented up to order 100. The main topic of this dissertation is the classification of generalized Hadamard matrices, difference matrices and Butson-type Hadamard matrices. Classification is performed by exhaustive computer search based on a general mathematical framework and computational algorithms that can be used with all types of orthogonal matrices investigated in this thesis. This computer-aided classification yields several new generalized Hadamard matrices, difference matrices and Butsontype Hadamard matrices. This thesis gives a full classification of all difference matrices and generalized Hadamard matrices with c columns defined over an abelian group of order g, where c <20 and 3 ≤ g ≤ 7.