Extremal Collections of $k$-Uniform Vectors

TL;DR

This paper establishes upper bounds on the number of specific weight vectors in matrices over finite fields, demonstrating the optimality of natural constructions and revealing their uniqueness under certain conditions.

Contribution

It provides new bounds on the extremal number of weight-$k$ vectors in matrices over finite fields, extending understanding of their structure and optimality.

Findings

Upper bounds on the number of weight-$k$ columns in rank-$r$ matrices.

Optimality of natural examples with $r$ rows for these bounds.

Proofs indicate uniqueness of these extremal configurations.

Abstract

We show any matrix of rank $r$ over $\mathbb{F}_q$ can have $\leq \binom{r}{k}(q-1)^k$ distinct columns of weight $k$ if $ k \leq O_q(\sqrt{\log r})$ (up to divisibility issues), and $\leq \binom{r}{k}(q-1)^{r-k}$ distinct columns of co-weight $k$ if $k \leq O_q(r^{2/3})$. 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.