Construction and Fast Decoding of Binary Linear Sum-Rank-Metric Codes
Sum-rank-metric codes have wide applications in multishot network coding and distributed storage. Linearized Reed-Solomon codes, sum-rank BCH codes and their Welch-Berlekamp decoding algorithms have been proposed and studied. In this paper, we construct binary linear sum-rank-metric codes in <inl...
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| Veröffentlicht in: | IEEE transactions on information theory Jg. 71; H. 12; S. 9319 - 9329 |
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01.12.2025
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| Abstract | Sum-rank-metric codes have wide applications in multishot network coding and distributed storage. Linearized Reed-Solomon codes, sum-rank BCH codes and their Welch-Berlekamp decoding algorithms have been proposed and studied. In this paper, we construct binary linear sum-rank-metric codes in <inline-formula> <tex-math notation="LaTeX">{\mathbf { F}}_{2}^{2 \times 2}\oplus {\mathbf { F}}_{2}^{2 \times 2} \oplus \cdots \oplus {\mathbf { F}}_{2}^{2 \times 2} </tex-math></inline-formula> from BCH, Goppa and additive quaternary codes. A reduction of decoding of binary sum-rank-metric codes to decoding of Hamming metric codes is given. Fast decoding algorithms of BCH-type and Goppa-type binary linear sum-rank-metric codes in <inline-formula> <tex-math notation="LaTeX">{\mathbf { F}}_{2}^{2 \times 2}\oplus {\mathbf { F}}_{2}^{2 \times 2} \oplus \cdots \oplus {\mathbf { F}}_{2}^{2 \times 2} </tex-math></inline-formula> with the block length <inline-formula> <tex-math notation="LaTeX">\ell </tex-math></inline-formula>, which are better than these sum-rank BCH codes, are presented. These fast decoding algorithms for BCH-type and Goppa-type binary linear sum-rank-metric codes need at most <inline-formula> <tex-math notation="LaTeX">O(\ell ^{2}) </tex-math></inline-formula> operations in the field <inline-formula> <tex-math notation="LaTeX">{\mathbf { F}}_{4} </tex-math></inline-formula>. Asymptotically good sequences of quadratic-time encodable and decodable binary linear sum-rank-metric codes with the matrix size <inline-formula> <tex-math notation="LaTeX">\times 2 </tex-math></inline-formula> are constructed from Goppa codes. |
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| AbstractList | Sum-rank-metric codes have wide applications in multishot network coding and distributed storage. Linearized Reed-Solomon codes, sum-rank BCH codes and their Welch-Berlekamp decoding algorithms have been proposed and studied. In this paper, we construct binary linear sum-rank-metric codes in <inline-formula> <tex-math notation="LaTeX">{\mathbf { F}}_{2}^{2 \times 2}\oplus {\mathbf { F}}_{2}^{2 \times 2} \oplus \cdots \oplus {\mathbf { F}}_{2}^{2 \times 2} </tex-math></inline-formula> from BCH, Goppa and additive quaternary codes. A reduction of decoding of binary sum-rank-metric codes to decoding of Hamming metric codes is given. Fast decoding algorithms of BCH-type and Goppa-type binary linear sum-rank-metric codes in <inline-formula> <tex-math notation="LaTeX">{\mathbf { F}}_{2}^{2 \times 2}\oplus {\mathbf { F}}_{2}^{2 \times 2} \oplus \cdots \oplus {\mathbf { F}}_{2}^{2 \times 2} </tex-math></inline-formula> with the block length <inline-formula> <tex-math notation="LaTeX">\ell </tex-math></inline-formula>, which are better than these sum-rank BCH codes, are presented. These fast decoding algorithms for BCH-type and Goppa-type binary linear sum-rank-metric codes need at most <inline-formula> <tex-math notation="LaTeX">O(\ell ^{2}) </tex-math></inline-formula> operations in the field <inline-formula> <tex-math notation="LaTeX">{\mathbf { F}}_{4} </tex-math></inline-formula>. Asymptotically good sequences of quadratic-time encodable and decodable binary linear sum-rank-metric codes with the matrix size <inline-formula> <tex-math notation="LaTeX">\times 2 </tex-math></inline-formula> are constructed from Goppa codes. |
| Author | Qi, Yanfeng Chen, Hao Cheng, Zhiqiang |
| Author_xml | – sequence: 1 givenname: Hao orcidid: 0000-0002-4558-8982 surname: Chen fullname: Chen, Hao email: haochen@jnu.edu.cn organization: College of Information Science and Technology, Jinan University, Guangzhou, Guangdong, China – sequence: 2 givenname: Zhiqiang surname: Cheng fullname: Cheng, Zhiqiang email: 2712468769@qq.com organization: College of Modern Economics and Management, Jiangxi University of Finance and Economics, Jiujiang, China – sequence: 3 givenname: Yanfeng orcidid: 0000-0003-1381-5471 surname: Qi fullname: Qi, Yanfeng email: qiyanfeng07@163.com organization: School of Science, Hangzhou Dianzi University, Hangzhou, China |
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| SubjectTerms | additive quaternary code Additives Codes Decoding decoding of sum-rank-metric code Encoding Geometry Hamming distances Hamming weight Linear codes Measurement quadratic-time encodable and decodable sum-rank-metric codes Reed-Solomon codes Sum-rank BCH code Welch-Berlekamp decoding algorithm |
| Title | Construction and Fast Decoding of Binary Linear Sum-Rank-Metric Codes |
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