On sets of vectors of a finite vector space in which every subset of basis size is a basis II

TL;DR

This paper proves the MDS conjecture for certain parameters, establishing bounds on the size of vector sets in finite fields where every subset of basis size is a basis, advancing understanding of maximum code lengths.

Contribution

It provides a proof of the MDS conjecture for $k \,\leq\, 2p-2$ and a short proof for $k \,\leq\, p$, extending known results in finite field coding theory.

Findings

Proves MDS conjecture for $k \leq 2p-2$

Provides a short proof for $k \leq p$ for all $q$

Establishes that such sets have size at most $q+1$

Abstract

This article contains a proof of the MDS conjecture for $k \leq 2p-2$. That is, that if $S$ is a set of vectors of ${\mathbb F}_q^k$ in which every subset of $S$ of size $k$ is a basis, where $q=p^h$, $p$ is prime and $q$ is not and $k \leq 2p-2$, then $|S| \leq q+1$. It also contains a short proof of the same fact for $k\leq p$, for all $q$.