A Survey of the Hadamard Maximal Determinant Problem

Abstract

In a celebrated paper of 1893, Hadamard established the maximal determinant theorem, which establishes an upper bound on the determinant of a matrix with complex entries of norm at most 1. His paper concludes with the suggestion that mathematicians study the maximum value of the determinant of an $n \times n$ matrix with entries in $\{ \pm 1\}$. This is the Hadamard maximal determinant problem. This survey provides complete proofs of the major results obtained thus far. We focus equally on upper bounds for the determinant (achieved largely via the study of the Gram matrices), and constructive lower bounds (achieved largely via quadratic residues in finite fields and concepts from design theory). To provide an impression of the historical development of the subject, we have attempted to modernise many of the original proofs, while maintaining the underlying ideas. Thus some of the proofs have the flavour of determinant theory, and some appear in print in English for the first time. We survey constructions of matrices in order $n \equiv 3 \mod 4$, giving asymptotic analysis which has not previously appeared in the literature. We prove that there exists an infinite family of matrices achieving at least 0.48 of the maximal determinant bound. Previously the best known constant for a result of this type was 0.34.