In this paper, a Roos like bound on the minimum distance for skew cyclic codes over a general field is provided. The result holds in the Hamming metric and in the rank metric. The proofs involve arithmetic properties of skew polynomials and an analysis of the rank of parity-check matrices. For the rank metric case, a way to arithmetically construct codes with a prescribed minimum rank distance, using the skew Roos bound, is also given. Moreover, some examples of MDS codes and MRD codes over finite fields are built, using the skew Roos bound.

中文翻译：

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在汉明和秩度量中偏斜循环码的Roos界

在本文中，提供了在一般字段上的偏斜循环码的最小距离上的Roos相似边界。结果保存在汉明度量标准和秩度量标准中。证明涉及偏多项式的算术性质以及对奇偶校验矩阵的秩的分析。对于等级度量的情况，还给出了一种使用偏斜Roos边界以算术方式构造具有规定最小等级距离的代码的方法。此外，使用偏斜Roos边界建立了有限域上MDS代码和MRD代码的一些示例。