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Özet (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.

Yazarların özeti; kaynağından alınmıştır. Journal of the European Mathematical Society, 2012 · DOI ↗