Construction and Fast Decoding of Binary Linear Sum-Rank-Metric Codes
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| Format: | Preprint |
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2023
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| _version_ | 1866913299044499456 |
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| author | Chen, Hao Qi, Yanfeng Cheng, Zhiqiang |
| author_facet | Chen, Hao Qi, Yanfeng Cheng, Zhiqiang |
| contents | 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 $$R_{sr}(δ_{sr}) \geq 1-\frac{1}{2}(H_4(\frac{4}{3}δ_{sr})+H_4(2δ_{sr})),$$ can be constructed from Goppa codes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_03619 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Construction and Fast Decoding of Binary Linear Sum-Rank-Metric Codes Chen, Hao Qi, Yanfeng Cheng, Zhiqiang Information Theory 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 $$R_{sr}(δ_{sr}) \geq 1-\frac{1}{2}(H_4(\frac{4}{3}δ_{sr})+H_4(2δ_{sr})),$$ can be constructed from Goppa codes. |
| title | Construction and Fast Decoding of Binary Linear Sum-Rank-Metric Codes |
| topic | Information Theory |
| url | https://arxiv.org/abs/2311.03619 |