Automorphism Ensemble Decoding of Reed-Muller Codes

TL;DR

This paper introduces a decoding architecture for Reed-Muller codes leveraging their automorphism groups, achieving near-ML performance with diverse decoders and analyzing theoretical limitations compared to polar codes.

Contribution

It presents a versatile automorphism-based decoding scheme for RM codes that generalizes existing methods and provides both empirical performance results and theoretical insights.

Findings

Near-ML performance for RM(3,7)-code at BLER of 10^{-3}

Decoding scheme is versatile, using various constituent decoders

Provides insights into automorphism subgroups and theoretical limits

Abstract

Reed-Muller (RM) codes are known for their good maximum likelihood (ML) performance in the short block-length regime. Despite being one of the oldest classes of channel codes, finding a low complexity soft-input decoding scheme is still an open problem. In this work, we present a versatile decoding architecture for RM codes based on their rich automorphism group. The decoding algorithm can be seen as a generalization of multiple-bases belief propagation (MBBP) and may use any polar or RM decoder as constituent decoders. We provide extensive error-rate performance simulations for successive cancellation (SC)-, SC-list (SCL)- and belief propagation (BP)-based constituent decoders. We furthermore compare our results to existing decoding schemes and report a near-ML performance for the RM(3,7)-code (e.g., 0.04 dB away from the ML bound at BLER of $10^{-3}$) at a competitive computational cost. Moreover, we provide some insights into the automorphism subgroups of RM codes and SC decoding and, thereby, prove the theoretical limitations of this method with respect to polar codes.