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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 $${\mathbb F}_q$$, with the property that every subset of size $$k$$ is a basis, is at most $$q+1$$, if $$k \leq p$$, and at most $$q+k-p$$, if $$q \geq k \geq p+1 \geq 4$$, where $$q=p^h$$ and $$p$$ is prime. Moreover, for $$k\leq p$$, 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 $$k \leq p$$ 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 $$n \geq r+2$$ is representable over $${\mathbb F}_p$$ if and only if $$n \leq p+1$$. 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.

##### MSC:
 15A03 Vector spaces, linear dependence, rank, lineability 05B35 Combinatorial aspects of matroids and geometric lattices 51E21 Blocking sets, ovals, $$k$$-arcs 94B05 Linear codes, general
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