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

Abstract

It is shown that the maximum size of a set $S$ of vectors of a $k$-dimensional vector space over ${\mathbb{F}}_q$, with the property that every subset of size $k$ is a basis, is at most $q+1$, if $k\le p$, and at most $q+k-p$, if $q\ge k\ge p+1\ge 4$, where $q=p^h$ and $p$ is prime. Moreover, for $k\le p$, the sets $S$ of maximum size are classified, generalising Beniamino Segre's “arc is a conic'' theorem.

These results have various implications. One such implication is that a $k\times (p+2)$ matrix, with $k\le p$ and entries from ${\mathbb{F}}_p$, has $k$ columns which are linearly dependent. Another is that the uniform matroid of rank $r$ that has a base set of size $n\ge r+2$ is representable over ${\mathbb{F}}_p$ if and only if $n\le p+1$. It also implies that the main conjecture for maximum distance separable codes is true for prime fields; that there are no maximum distance separable linear codes over ${\mathbb{F}}_p$, of dimension at most $p$, longer than the longest Reed-Solomon codes. The classification implies that the longest maximum distance separable linear codes, whose dimension is bounded above by the characteristic of the field, are Reed–Solomon codes.