A Partition Decoding for Reed-Solomon Codes Based on Partial Bit Reliability

Abstract

This study presents a partition decoding algorithm for an (mN, mK) binary image of an (N, K) Reed Solomon code over GF(2 m ). A proposed partition decoding algorithm includes several steps. Firstly we compute m's segmental reliability values of a received subvector of length N and determine which one with the least segmental reliability value. A permutation is performed on a binary generator matrix of an RS code and a received vector, which are then partitioned into two submatrices and two subvectors. The first subvector of length N(m - 1) associate with the first submatrix and the second subvector with the least segmental reliability value relates to the second submatrix. Secondly, an MLD algorithm based on the first submatrix is employed to decode the first subvector. Thirdly, an MLD algorithm based on a BCH generator matrix is employed to decode the second subvector. A codeword is finally outputted after performing the inverse permutation on a concatenation of code vectors decoded from these two decoding. The error coefficient and minimum Hamming distance of the code sequences generated in the first submatrix are fewer than those of a corresponding binary image. Simulation results show that at low and medium SNRs, the effect of error coefficient becomes more significant than that of minimum Hamming distance. Minimum Hamming distances and error coefficients of code sequences generated in the first submatrices and their corresponding binary images have been explored in this work. For (60,36,7)RS (b) , (155,125,7)RS (b) , (155,105,11)RS (b) and (889, 847,7))RS (b) being binary images of (15,9,7)RS, (31,25,7)RS, (31,21,11)RS and (127,121,7)RS codes respectively, with BPSK signaling over AWGN channels, the decoding performances of proposed partition decoding algorithm are a little poorer than those of MLD [10] by 1.0 to 1.4dB at BER 10 -5 , but better than those of GMD decoding [1] by 0.8 to 1.1 dB. For SNR of 5 dB, proposed partition decoding algorithm only takes 50% to 60% amount of bit operations of an MLD [10]. Under a constraint of decoding complexity, proposed partition decoding algorithm may be a solution to decode binary images of long RS codes, which provides superior performance to GMD decoding with much lower complexity than an MLD.