New Explicit Good Linear Sum-Rank-Metric Codes

Sum-rank-metric codes have wide applications in universal error correction and security in multishot network, space-time coding and construction of partial-MDS codes for repair in distributed storage. Fundamental properties of sum-rank-metric codes have been studied and some explicit or probabilistic constructions of good sum-rank-metric codes have been proposed. In this paper we propose three simple constructions of explicit linear sum-rank-metric codes. In finite length regime, numerous good linear sum-rank-metric codes from our construction are given. Some of them have better parameters than previous constructed sum-rank-metric codes. For example a lot of small block size better linear sum-rank-metric codes over ${\bf F}_q$ of the matrix size $2 \times 2$ are constructed for $q=2, 3, 4$. Asymptotically our constructed sum-rank-metric codes are closing to the Gilbert-Varshamov-like bound for the sum-rank-metric codes for some parameters. Our construction also leads to one-weight or few nonzero weight linear sum-rank-metric codes with large minimum sum-rank distances. Hamming anticodes with large minimum pair distances and smaller diameters can be used to get sum-rank-metric codes with large minimum sum-rank distances.