Determinants of binary matrices achieve every integral value up to Ω(2n/n)

Abstract

This work shows that the smallest natural number dn that is not the determinant of some n×n binary matrix is at least c2n/n for c=1/201. That same quantity naturally lower bounds the number of distinct integers Dn which can be written as the determinant of some n×n binary matrix. This asymptotically improves the previous result of dn=Ω(1.618n) and slightly improves the previous result of Dn≥2n/g(n) for a particular g(n)=ω(n2) function.