Reed-Solomon Coding Algorithms Based on Reed-Muller Transform for Any Number of Parities
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. Th...
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| Published in | IEEE transactions on computers Vol. 72; no. 9; pp. 2677 - 2688 |
|---|---|
| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
New York
IEEE
01.09.2023
The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0018-9340 1557-9956 |
| DOI | 10.1109/TC.2023.3262922 |
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| Abstract | 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. |
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| AbstractList | 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. 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 [Formula Omitted] XORs per data bit with [Formula Omitted] increasing, where [Formula Omitted] and [Formula Omitted] 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. |
| Author | Yu, Leilei Li, Zhengrui Lin, Sian-Jheng Hou, Hanxu |
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| Snippet | Based on the Reed-Muller (RM) transform, this paper proposes a Reed-Solomon (RS) encoding/erasure decoding algorithm for any number of parities. Specifically,... |
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| SubjectTerms | 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 |
| Title | Reed-Solomon Coding Algorithms Based on Reed-Muller Transform for Any Number of Parities |
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