Lower bounds on maximal determinants of binary matrices via the probabilistic method

TL;DR

This paper establishes new probabilistic lower bounds on the maximal determinants of binary matrices, relating them to Hadamard matrices, and extends previous results to broader cases.

Contribution

It introduces several novel lower bounds on the ratio of maximal determinants, generalizing prior work and utilizing the probabilistic method for broader matrix sizes.

Findings

New lower bounds on ${ m R}(n)$ in terms of $d$

Asymptotically sharper bounds involving exponential terms

Results applicable for large $n$ and small $d$, including the case $d o 0$

Abstract

Let $D(n)$ be the maximal determinant for $n \times n$ $\{\pm 1\}$-matrices, and ${\mathcal R}(n) = D(n)/n^{n/2}$ be the ratio of $D(n)$ to the Hadamard upper bound. We give several new lower bounds on ${\mathcal R}(n)$ in terms of $d$, where $n = h+d$, $h$ is the order of a Hadamard matrix, and $h$ is maximal subject to $h \le n$. A relatively simple bound is \[{\mathcal R}(n) \ge \left(\frac{2}{\pi e}\right)^{d/2} \left(1 - d^2\left(\frac{\pi}{2h}\right)^{1/2}\right) \;\text{ for all }\; n \ge 1.\] An asymptotically sharper bound is \[{\mathcal R}(n) \ge \left(\frac{2}{\pi e}\right)^{d/2} \exp\left(d\left(\frac{\pi}{2h}\right)^{1/2} + \; O\left(\frac{d^{5/3}}{h^{2/3}}\right)\right).\] We also show that \[{\mathcal R}(n) \ge \left(\frac{2}{\pi e}\right)^{d/2}\] if $n \ge n_0$ and $n_0$ is sufficiently large, the threshold $n_0$ being independent of $d$, or for all $n\ge 1$ if $0 \le d \le 3$ (which would follow from the Hadamard conjecture). The proofs depend on the probabilistic method, and generalise previous results that were restricted to the cases $d=0$ and $d=1$.