Row Space Cardinalities

AbstractLet ${\cal B}_n$ be the set of all $n\times n$ Boolean matrices. Let R(A) denote the row space of $A\in{\cal B}_n$, let ${\cal R}_n=\{r \mid r={\rm r}(A),\ A\in {\cal B}_n \}$, and let $a_n=\min\{q\ge 1\mid q\notin {\cal R}_n\}$. By extensive computation we found that ${\cal R}_9\cap[1,256]=[1,190]\cup [192,204]\cup\{206\}\cup[208,212]\cup\{214,216,220\}\cup\,[224,228]\cup\{230,232,236,240,248,256\},$ and therefore $a_9=191$. Furthermore, $a_n\ge 5{\sqrt[11]{336}\,}^n$ for $n\ge 31$. We proved that if $n\ge 7$, then the set ${\cal R}_n\cap(2^{n-2}+2^{n-3},2^{n-1}]$ contains at least $n^2-7n+14+\frac{1}{24}\left((n-8)(n-10)(2n-15)+3(n\bmod 2)\right)$ elements.