Combinatorics of Complex Maximal Determinant Matrices

This doctoral thesis covers several topics related to the construction and study of maximal determinant matrices with complex entries. The first three chapters are devoted to number-theoretic tools to prove the non-solvability of Gram matrix equations over certain fields, with a focus on combinatorial applications. Chapter 4 gives a survey on Butson-type Hadamard matrices, and shows an improved lower bound on primes $p$ for the existence of $BH(12p, p)$ matrices. Chapter 5 contains the main contributions of the thesis, where the maximal determinant problem for matrices over the m-th roots of unity is discussed, and where new upper and lower bounds, as well as constructions at small orders, are given. Chapter 6 studies maximal determinant matrices over association schemes. Chapter 7 gives an application of design theory to privacy in communications, and it is connected to the rest of the thesis by the use of the theory of quadratic forms.