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

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.