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

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 xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\lfloor \mathrm{lg} T \rfloor + 1$</tex-math></inline-formula> XORs per data bit with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> increasing, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$T$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></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.