Decoding Reed-Muller Codes with Successive Codeword Permutations

A novel recursive list decoding (RLD) algorithm for Reed-Muller (RM) codes based on successive permutations (SP) of the codeword is presented. A low-complexity SP scheme applied to a subset of the symmetry group of RM codes is first proposed to carefully select a good codeword permutation on the fly. Then, the proposed SP technique is integrated into an improved RLD algorithm that initializes different decoding paths with random codeword permutations, which are sampled from the full symmetry group of RM codes. Finally, efficient latency and complexity reduction schemes are introduced that virtually preserve the error-correction performance of the proposed decoder. Simulation results demonstrate that at the target frame error rate of $10^{-3}$ for the RM code of size $256$ with $163$ information bits, the proposed decoder reduces $33\%$ of the computational complexity and $60\%$ of the decoding latency of a simplified successive-cancellation decoder with fast Hadamard transform (SSC-FHT) that uses $128$ permutations from the full symmetry group of RM codes, at the cost of negligible error-correction performance loss and memory consumption overhead.