Reed-Solomon Coding Algorithms Based on Reed-Muller Transform for Any Number of Parities

• Published in: IEEE transactions on computers Vol. 72; no. 9; pp. 2677 - 2688
• Main Authors: Yu, Leilei, Lin, Sian-Jheng, Hou, Hanxu, Li, Zhengrui
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.09.2023; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Codes; Coding; computational complexity; Computational efficiency; Computing time; Decoding; Encoding; Galois fields; Linear equations; Mathematical analysis; Mathematical models; Matrix decomposition; Reed-Muller transform; Reed-Solomon code; storage erasure codes; Transforms; Vandermonde matrix
• ISSN: 0018-9340; 1557-9956
• DOI: 10.1109/TC.2023.3262922

Based on the Reed-Muller (RM) transform, this paper proposes a Reed-Solomon (RS) encoding/erasure decoding algorithm for any number of parities. Specifically, we first generalize the previous RM-based syndrome calculation, which allows only up to seven parities, to support any number of parities. Then we propose a general encoding/erasure decoding algorithm. The proposed encoding algorithm eliminates the operations in solving linear equations, and this improves the computational efficiency of existing RM-based RS algorithms. In terms of erasure decoding, this paper employs the generalized RM-based syndrome calculation and lower-upper (LU) decomposition to accelerate the computational efficiency. Analysis shows that the proposed encoding/erasure decoding algorithm approaches the complexity of <inline-formula><tex-math notation="LaTeX">\lfloor \lg T \rfloor + 1</tex-math> <mml:math><mml:mrow><mml:mo>⌊</mml:mo><mml:mo form="prefix">lg</mml:mo><mml:mi>T</mml:mi><mml:mo>⌋</mml:mo><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="yu-ieq1-3262922.gif"/> </inline-formula> XORs per data bit with <inline-formula><tex-math notation="LaTeX">N</tex-math> <mml:math><mml:mi>N</mml:mi></mml:math><inline-graphic xlink:href="yu-ieq2-3262922.gif"/> </inline-formula> increasing, where <inline-formula><tex-math notation="LaTeX">T</tex-math> <mml:math><mml:mi>T</mml:mi></mml:math><inline-graphic xlink:href="yu-ieq3-3262922.gif"/> </inline-formula> and <inline-formula><tex-math notation="LaTeX">N</tex-math> <mml:math><mml:mi>N</mml:mi></mml:math><inline-graphic xlink:href="yu-ieq4-3262922.gif"/> </inline-formula> denote the number of parities and codeword length respectively. To highlight the advantage of the proposed RM-based algorithms, the implementations with Single Instruction Multiple Data (SIMD) technology are provided. Simulation results show that the proposed algorithms are competitive, as compared with other cutting-edge implementations.