General lower bounds on maximal determinants of binary matrices

TL;DR

This paper establishes improved lower bounds on the maximum determinants of n by n {+1,-1}-matrices, advancing theoretical understanding in combinatorial matrix theory with implications for Hadamard matrices.

Contribution

It provides new, tighter lower bounds on maximal determinants of {+1,-1}-matrices, including bounds independent of the Hadamard conjecture, and offers a novel proof related to minors of Hadamard matrices.

Findings

Improved lower bounds on maximal determinants for various n.

Bounds that surpass previous results for certain congruence classes.

A new proof using Jacobi's identity for minors of Hadamard matrices.

Abstract

We give general lower bounds on the maximal determinant of n by n {+1,-1}-matrices, both with and without the assumption of the Hadamard conjecture. Our bounds improve on earlier results of de Launey and Levin (2010) and, for certain congruence classes of n mod 4, those of Koukouvinos, Mitrouli and Seberry (2000). In an Appendix we give a new proof, using Jacobi's determinant identity, of a result of Sz\"oll\H{o}si (2010) on minors of Hadamard matrices.