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首页> 外文期刊>Journal of the European Mathematical Society: JEMS >On sets of vectors of a finite vector space in which every subset of basis size is a basis
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On sets of vectors of a finite vector space in which every subset of basis size is a basis

机译：在有限向量空间的向量集上，其中基础大小的每个子集都是基础

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摘要

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.
机译：结果表明，在F_q上，k维向量空间的向量集S的最大大小为k的每个子集都是基础，如果k≤p，则最大q等于1。如果q≥k≥p + 1≥4，则大多数q + k-p，其中q = ph且p为质数。此外，对于k≤p，归纳Beniamino Segre的“弧是圆锥”定理，对最大尺寸的集合S进行分类。这些结果具有各种含义。一个这样的暗示是，一个k×（p + 2）矩阵，其中k≤p，并且来自F _p的条目，具有k个与线性相关的列。另一个是，当且仅当n≤p +1时，才能在F _p上表示具有大小为n≥r + 2的基集的秩r的均匀拟阵。这也意味着最大距离可分离码的主要猜想对于素数场是正确的。在F _p上没有最大距离可分离的线性码，最大维数为p，最长的Reed-Solomon码更长。该分类意味着最长尺寸的最大距离可分离线性码是Reed-Solomon码，其尺寸在上方受字段的特性限制。

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