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

### Simeon Ball

Universitat Politécnica de Catalunya, Barcelona, Spain

## Abstract

It is shown that the maximum size of a set $S$ of vectors of a $k$-dimensional vector space over $F_{q}$, with the property that every subset of size $k$ is a basis, is at most $q+1$, if $k≤p$, and at most $q+k−p$, if $q≥k≥p+1≥4$, where $q=p_{h}$ and $p$ is prime. Moreover, for $k≤p$, 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 $k×(p+2)$ matrix, with $k≤p$ and entries from $F_{p}$, has $k$ columns which are linearly dependent. Another is that the uniform matroid of rank $r$ that has a base set of size $n≥r+2$ is representable over $F_{p}$ if and only if $n≤p+1$. 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 $F_{p}$, 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.

## Cite this article

Simeon Ball, On sets of vectors of a finite vector space in which every subset of basis size is a basis. J. Eur. Math. Soc. 14 (2012), no. 3, pp. 733–748

DOI 10.4171/JEMS/316