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This article contains a proof of the MDS conjecture for $k \leq 2p-2$. Thatis, that if $S$ is a set of vectors of ${\mathbb F}_q^k$ in which every subsetof $S$ 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 samefact for $k\leq p$, for all $q$.

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