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

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 $c{2}^n\mathrm{/}n$ for $c=1\mathrm{/}201$. 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 $d_n=\mathrm{\Omega }({1.618}^n)$ and slightly improves the previous result of $D_n\ge 2^n/g(n)$foraparticular$g(n)=\omega(n^2)$ function.