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ON LARGE SUBSETS 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 Fq, with the property that every subset of size k is a basis, is at most q + 1, if k is bounded above by the characteristic p, and at most q + k − p, if k ≥ 4 is larger than the characteristic. Moreover, for 3 ≤ k ≤ p, the sets S of maximum size are classified. These results have various implications. One such implication is that a k × (p + 2) matrix, with k ≤ p and entries from Fp, 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 Fp 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 Fp, 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.