秩度量中码的性质
Properties of codes in rank metric
摘要：
研究了有限域上秩度量和秩度量下码的性质。我们证明了在秩度量完美码是不存在的。我们得到的存在界是等价的Gilbert-Varshamov界在汉明度量。研究了满足GV的码的最小秩距离的渐近性态。本文导出了随机码和随机$\F{q}$ -线性码的最小秩距离的概率分布。我们给出了它们的平均最小秩距离的一个渐近等价式，并证明了随机$\F{q}$ -线性码对于秩度量是GV有界的。我们发现，覆盖密度的最佳代码的码字可以被看作是方阵的下限由一个函数只依赖于代码的纠错能力。证明了特征为2的域上存在秩度量的拟完美码。
英文摘要：
We study properties of rank metric and codes in rank metric over finite fields. We show that in rank metric perfect codes do not exist. We derive an existence bound that is the equivalent of the Gilbert--Varshamov bound in Hamming metric. We study the asymptotic behavior of the minimum rank distance of codes satisfying GV. We derive the probability distribution of minimum rank distance for random and random $\F{q}$-linear codes. We give an asymptotic equivalent of their average minimum rank distance and show that random $\F{q}$-linear codes are on GV bound for rank metric. We show that the covering density of optimum codes whose codewords can be seen as square matrices is lower bounded by a function depending only on the error-correcting capability of the codes. We show that there are quasi-perfect codes in rank metric over fields of characteristic 2.
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