  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.
