On Decoding of Reed-Muller Codes Using a Local Graph Search

TL;DR

This paper introduces a new iterative decoding algorithm for Reed-Muller codes that uses a graph-based local search approach, achieving near-optimal performance and outperforming existing algorithms for certain code lengths.

Contribution

The paper proposes a novel graph-based local search decoding method for Reed-Muller codes, improving decoding performance and computational efficiency over existing algorithms.

Findings

Approaches maximum likelihood decoding performance for certain code lengths.

Outperforms state-of-the-art decoding algorithms at similar complexity.

Effective for Reed-Muller codes of length less than 1024 and 4096 for second-order codes.

Abstract

We present a novel iterative decoding algorithm for Reed-Muller (RM) codes, which takes advantage of a graph representation of the code. Vertices of the considered graph correspond to codewords, with two vertices 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 uses a greedy local search to find a node optimizing a metric, e.g. the correlation between the received vector and the corresponding codeword. In addition, the cyclic redundancy check can be used to terminate the search as soon as a valid codeword is found, leading to an improvement 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 approaches the performance of maximum likelihood decoding for RM codes of length less than 1024 and for the second-order RM codes of length less than 4096. Moreover, it is demonstrated that the considered decoding approach outperforms state-of-the-art decoding algorithms of RM codes with similar computational complexity for a wide range of block lengths and rates.