Extremal Collections of $k$-Uniform Vectors

We show any matrix of rank $r$ over ${\mathbb{F}}_q$ can have $\leq \binom{r}{k}(q-1)^k$distinctcolumnsofweight$k$if$ k \leq O_q(\sqrt{\log r})$(uptodivisibilityissues),and$\leq \binom{r}{k}(q-1)^{r-k}$ distinct columns of co-weight $k$ if $k\le O_q(r^{2\mathrm{/}3})$. This shows the natural examples consisting of only $r$ rows are optimal for both, and the proofs will recover some form of uniqueness of these examples in all cases.