Construction and Fast Decoding of Binary Linear Sum-Rank-Metric Codes - Chen, Hao; Cheng, Zhiqiang; Qi, Yanfeng (IEEE transactions on information theory, 2025)
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 { F}_{2}^{2 × 2}\oplus { F}_{2}^{2 × 2} \oplus … \oplus { F}_{2}^{2 × 2} 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 { F}_{2}^{2 × 2}\oplus { F}_{2}^{2 × 2} \oplus … \oplus { F}_{2}^{2 × 2} with the block length \ell , 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 O(\ell ^{2}) operations in the field { F}_{4} . Asymptotically good sequences of quadratic-time encodable and decodable binary linear sum-rank-metric codes with the matrix size × 2 are constructed from Goppa codes.