 Prime-power modular intersection bound

Let $m=p^a$ and let an $r$-uniform set family have no distinct pair whose intersection is congruent to $r$ modulo $m$. Its size is bounded as displayed; the customary nontrivial form $\left(\genfrac{}{}{0ex}{}{n}{m-1}\right)$ requires $m-1\le r$. The Lucas theorem makes $\left(\genfrac{}{}{0ex}{}{z-r+m-1}{m-1}\right)$ a residue detector modulo $p$, equal to one exactly when $z\equiv r(\mathrm{mod}m)$. Its Boolean evaluation functions have degree at most $m-1$ and a diagonal evaluation matrix on the family. The low-degree evaluation rank on a uniform layer supplies the sharp dimension bound. Without the size qualification, three singletons with $n=3,r=1,m=4$ contradict the customary printed form, since $3>\left(\genfrac{}{}{0ex}{}{3}{3}\right)$.