We provide an algebraic description for sum-rank metric codes, as quotient space of a skew polynomial ring. This approach generalizes at the same time the skew group algebra setting for rank-metric codes and the polynomial setting for codes in the Hamming metric. This allows to construct twisted linearized Reed-Solomon codes, a new family of maximum sum-rank distance codes extending at the same time Sheekey's twisted Gabidulin codes in the rank metric and twisted Reed-Solomon codes in the Hamming metric. Furthermore, we provide an analogue in the sum-rank metric of Trombetti-Zhou construction, which also provides a family of maximum sum-rank distance codes. As a byproduct, in the extremal case of the Hamming metric, we obtain a new family of additive MDS codes over quadratic fields.
翻译：我们用代数描述成一模一样的公元体代码,作为一个扭曲的多元圆环的商数空间。这个方法同时将标准码的折叠组代数设置和Hamming 标准中代码的多元代数设置加以概括。这可以构建扭曲的线性Reed-Solomon代码,这是一个由最高和最高距离代码组成的新体系,与Sheekey的扭曲的公元体代码和Hamming 标准中扭曲的Reed-Solomon代码同时延伸。此外,我们提供了Trombetti-Zhou建筑总和公元的类比喻,它也提供了最高和最高距离代码的组合。作为副产品,在Hamming 标准中,我们获得了一个在四极域的添加式MDS代码的新组合。
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