Codes in the sum-rank metric have received many attentions in recent years, since they have wide applications in the multishot network coding, the space-time coding and the distributed storage. Fundamental bounds, some explicit or probabilistic constructions of sum-rank codes and their decoding algorithms have been developed in previous papers. In this paper, we construct covering codes in the sum-rank metric from covering codes in the Hamming metric. Then some upper bounds on sizes of covering codes in the sum-rank metric are presented. Block length functions of covering codes in the sum-rank metric are also introduced and studied. As applications of our upper bounds on covering codes in the sum-rank metric and block length functions, several strong Singleton-like bounds on sum-rank codes are proposed and proved. These strong Singleton-like bounds are much stronger than the Singleton-like bound for sum-rank codes, when block lengths are larger and minimum sum-rank distances are small. An upper bound on sizes of list-decodable codes in the sum-rank metric is also given, which leads to an asymptotic bound on list-decodability of sum-rank codes. We also give upper bounds on block lengths of general MSRD codes.

翻译：近年来，和-秩度量下的编码因其在多跳网络编码、空时编码和分布式存储中的广泛应用而备受关注。已有文献发展了和-秩码的基本界、显式或概率构造方法及译码算法。本文通过汉明度量下的覆盖码构造了和-秩度量下的覆盖码，并给出了和-秩度量下覆盖码规模的一些上界。同时引入并研究了和-秩度量下覆盖码的块长函数。作为覆盖码规模上界与块长函数的应用，我们提出并证明了和-秩码的若干强类Singleton界。当块长较大且最小和-秩距离较小时，这些强类Singleton界显著强于和-秩码的类Singleton界。此外，本文还给出了和-秩度量下列表可译码规模的上界，该上界导出了和-秩码列表可译性的渐进界。最后，我们得到了通用MSRD码的块长上界。