Extremal Collections of $k$-Uniform Vectors

Joseph Briggs, Wesley Pegden

We show any matrix of rank $r$ over ${\mathbb{F}}_q$ can have $\le \left(\genfrac{}{}{0ex}{}{r}{k}\right)(q-1{)}^k$ distinct columns of weight $k$ if $k\le O_q(\sqrt{\log r})$ (up to divisibility issues), a…