Association schemes and mutually unbiased Hadamard matrices

Abstract

This thesis is an examination of the theory of association schemes. A part of this thesis focuses on the relation between association schemes and different combinatorial objects. Special attention is paid to the notion of Bose-Mesner algebra of an association scheme, which leads us to eigenmatrices, Krein matrices, and intersection matrices. It is shown that if an association scheme is imprimitive, it can be used to generate a quotient association scheme. Different constructions are used to generate association schemes. This thesis explains how to generate a new class of association schemes from mutually unbiased Bushtype Hadamard matrices (abbreviated as MUBH). This class of association schemes leads to an upper bound on the number of mutually unbiased Bush-type Hadamard matrices. Lastly, the existence of this class of association schemes results in the existence of sets of MUBH