Determinants of binary matrices achieve every integral value up to $\Omega(2^n/n)$

TL;DR

This paper establishes a new lower bound on the smallest non-determinant of binary matrices, showing it grows at least as fast as a constant times 2^n/n, which improves previous bounds and enhances understanding of the determinants achievable by binary matrices.

Contribution

The paper provides the first asymptotic lower bound of order 2^n/n for the smallest unattainable determinant of binary matrices, improving prior exponential bounds.

Findings

The minimal unattainable determinant grows at least as c*2^n/n.

The number of attainable determinants D_n is asymptotically lower bounded by 2^n.

Previous exponential bounds are improved to polynomial-exponential bounds.

Abstract

This work shows that the smallest natural number $d_n$ that is not the determinant of some $n\times n$ binary matrix is at least $c\,2^n/n$ for $c=1/201$. That same quantity naturally lower bounds the number of distinct integers $D_n$ which can be written as the determinant of some $n\times n$ binary matrix. This asymptotically improves the previous result of $d_n=\Omega(1.618^n)$ and slightly improves the previous result of $D_n\ge 2^n/g(n)$ for a particular $g(n)=\omega(n^2)$ function.