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.

翻译：和秩度量码在多重信道通用纠错与安全、时空编码以及分布式存储修复用部分MDS码构造中具有广泛应用。目前已有关于和秩度量码基本性质的研究，并提出了若干显式或概率性构造优和秩度量码的方法。本文提出三种简单构造显式线性秩度量码的方案。在有限长范围内，我们通过所提构造给出了大量优线性秩度量码，其中多数参数优于此前构造的和秩度量码。例如，针对$q=2,3,4$的有限域${\bf F}_q$上$2 \times 2$矩阵尺寸，构造了大量小分组长度的优线性和秩度量码。渐近特性表明，所构造的和秩度量码在某些参数下趋近于和秩度量码的类Gilbert-Varshamov界。最后，我们在任意有限域${\bf F}_q$上构造了线性MSRD码，其矩阵尺寸$n_1>n_2>\cdots>n_t$满足$n_i \geq n_{i+1}^2+\cdots+n_t^2$（$i=1,2,\ldots,t-1$），且对于任意给定最小和秩距离，这些线性MSRD码的分组长度$t$及参数$N=n_1+\cdots+n_t$不受有限域${\bf F}_q$规模的限制。我们将证明，本文构造的线性和秩度量码的译码可简化为汉明度量下的译码。