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 of vectors of a -dimensional vector space over , with the property that every subset of size is a basis, is at most , if , and at most , if , where and is prime. Moreover, for , the sets 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 matrix, with and entries from , has columns which are linearly dependent. Another is that the uniform matroid of rank that has a base set of size is representable over if and only if . 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 , of dimension at most , 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.
|15A03||Vector spaces, linear dependence, rank|
|05B35||Matroids, geometric lattices (combinatorics)|
|51E21||Blocking sets, ovals, -arcs|
|94B05||General theory of linear codes|