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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Bibliographic Details
Published in:IEEE transactions on information theory Vol. 71; no. 12; pp. 9319 - 9329
Main Authors: Chen, Hao, Cheng, Zhiqiang, Qi, Yanfeng
Format: Journal Article
Language:English
Published: IEEE 01.12.2025
Subjects:
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
ISSN:0018-9448, 1557-9654
Online Access:Get full text
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Summary: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.
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2025.3619572