New Explicit Good Linear Sum-Rank-Metric Codes
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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 F_q of the matrix size 2 × 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 F_q with various matrix sizes n_1>n_2>⋯>n_t satisfying n_i ≥ n_i+1^2+⋯+n_t^2 , i=1, 2, …, t-1, for any given minimum sum-rank distance. There is no restriction on the block lengths t and parameters N=n_1+⋯+n_t of these linear MSRD codes from the sizes of the fields F_q.
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