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On sets of vectors of a finite vector space in which every subset of basis size is a basis. (English) Zbl 1241.15002
Summary: 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 $k\le p$, 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 $k×\left(p+2\right)$ matrix, with $k\le p$ 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 $n\ge r+2$ is representable over ${𝔽}_{p}$ 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 ${𝔽}_{p}$, 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.

##### MSC:
 15A03 Vector spaces, linear dependence, rank 05B35 Matroids, geometric lattices (combinatorics) 51E21 Blocking sets, ovals, $k$-arcs 94B05 General theory of linear codes