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. Most 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 on sum-rank-metric codes for some parameters. Finally we construct a linear MSRD code over an arbitrary finite field ${\bf F}_q$ with various matrix sizes $n_1>n_2>\cdots>n_t$ satisfying $n_i \geq n_{i+1}^2+\cdots+n_t^2$ , $i=1, 2, \ldots, t-1$, for any given minimum sum-rank distance. There is no restriction on the block lengths $t$ and parameters $N=n_1+\cdots+n_t$ of these linear MSRD codes from the sizes of the fields ${\bf F}_q$. We will show that the decoding of linear sum-rank-metric codes constructed in this paper can be reduced to the decoding in the Hamming metric.