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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 kp, and at most q+k-p, if qkp+14, where q=p h and p is prime. Moreover, for kp, 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 kp 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 nr+2 is representable over 𝔽 p if and only if np+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:
15A03Vector spaces, linear dependence, rank
05B35Matroids, geometric lattices (combinatorics)
51E21Blocking sets, ovals, k-arcs
94B05General theory of linear codes