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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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