On Decoding of Reed-Muller Codes Using Local Graph Search

We present a novel iterative decoding algorithm for Reed-Muller (RM) codes, which takes advantage of a graph representation of the code. Vertexes of the considered graph correspond to codewords, with two vertexes being connected by an edge if and only if the Hamming distance between the corresponding codewords equals the minimum distance of the code. The algorithm starts from a random node and uses a greedy local search to find a node with the best heuristic, e.g. Euclidean distance between the received vector and the corresponding codeword. In addition, the cyclic redundancy check can be used to terminate the search as soon as the correct codeword is found, leading to a decrease in the average computational complexity of the algorithm. Simulation results for both binary symmetric channel and additive white Gaussian noise channel show that the presented decoder achieves the performance of maximum likelihood decoding for RM codes of length up to 512 and for the second order RM codes of length up to 2048. Moreover, it is demonstrated that the considered decoding approach significantly outperforms state-of-the-art decoding algorithms of RM codes with similar computational complexity for a large number of cases.