Construction and Fast Decoding of Binary Linear Sum-Rank-Metric Codes
| Autor: | Chen, Hao, Qi, Yanfeng, Cheng, Zhiqiang |
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| Rok vydání: | 2023 |
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| Druh dokumentu: | Working Paper |
| Popis: | Sum-rank-metric codes have wide applications in the multishot network coding and the distributed storage. Linearized Reed-Solomon codes, sum-rank BCH codes and their Welch-Berlekamp type decoding algorithms were proposed and studied. They are sum-rank versions of Reed-Solomon codes and BCH codes in the Hamming metric. In this paper, we construct binary linear sum-rank-metric codes of the matrix size $2 \times 2$, from BCH, Goppa and additive quaternary Hamming metric codes. Larger sum-rank-metric codes than these sum-rank BCH codes of the same minimum sum-rank distances are obtained. Then a reduction of the decoding in the sum-rank-metric to the decoding in the Hamming metric is given. Fast decoding algorithms of BCH and Goppa type binary linear sum-rank-metric codes of the block length $t$ and the matrix size $2 \times 2$, which are better than these sum-rank BCH codes, are presented. These fast decoding algorithms for BCH and Goppa type binary linear sum-rank-metric codes of the matrix size $2 \times 2$ need at most $O(t^2)$ operations in the field ${\bf F}_4$. Asymptotically good sequences of quadratic-time encodable and decodable binary linear sum-rank-metric codes of the matrix size $2 \times 2$ satisfying $$R_{sr}(\delta_{sr}) \geq 1-\frac{1}{2}(H_4(\frac{4}{3}\delta_{sr})+H_4(2\delta_{sr})),$$ can be constructed from Goppa codes. Comment: 37 pages, submitted |
| Databáze: | arXiv |
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