On sets of vectors of a finite vector space in which every subset of basis size is a basis

Abstract

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 ≤ p, and at most q + k - p if q ≥ k ≥ p + 1 ≥ 4, where q = p h and p is prime. Moreover, for k ≤ 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 × (p + 2) matrix, with k ≤ 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 ≥ r + 2 is representable over F p if and only if n ≤ 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. © European Mathematical Society 2012.