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. In this paper, by constructing covering codes in the sum-rank metric from covering codes in the Hamming metric, we derive new upper bounds on sizes, the covering radii and the block length functions of codes in the sum-rank metric. As applications, we present several strong Singleton-like bounds that are tighter than the classical Singleton-like bound when block lengths are large. In addition, we give the explicit constructions of the distance-optimal sum-rank codes of matrix sizes $s\times s$ and $2\times 2$ with minimum sum-rank distance four respectively by using cyclic codes in the Hamming metric. More importantly, we present an infinite families of quasi-perfect $q$-ary sum-rank codes with matrix sizes $2\times m$. Furthermore, we construct almost MSRD codes with larger block lengths and demonstrate how the Plotkin sum can be used to give more distance-optimal sum-rank codes.

翻译：和秩度量码近年来受到广泛关注，因其在多跳网络编码、空时编码和分布式存储中具有广泛应用。本文通过从汉明度量覆盖码构造和秩度量覆盖码，推导了和秩度量码在码本大小、覆盖半径和分组长度函数上的新上界。作为应用，我们提出了若干强Singleton类界，这些界在分组长度较大时比经典的Singleton类界更紧。此外，我们分别利用汉明度量下的循环码，给出了矩阵尺寸为$s\times s$和$2\times 2$、最小和秩距离为四的距离最优和秩码的显式构造。更重要的是，我们提出了矩阵尺寸为$2\times m$的拟完美$q$元和秩码的无限族。此外，我们构造了具有更大分组长度的近似MSRD码，并展示了如何利用Plotkin和来构造更多距离最优的和秩码。