Strong Singleton-Like Bounds, Quasi-Perfect Codes and Distance-Optimal Codes in the Sum-Rank Metric

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.