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 {\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.

The authors' abstract, as published at the source. Journal of the European Mathematical Society, 2012 · DOI ↗