Recursive Decoding of Reed-Muller Codes Starting With the Higher-Rate Constituent Code

TL;DR

This paper introduces a novel recursive decoding approach for Reed-Muller codes that decodes the higher-rate constituent code first, improving performance and efficiency, especially for longer codes, approaching maximum-likelihood decoding.

Contribution

The paper proposes decoding the higher-rate code first in recursive RM decoding, enabling efficient permutation-based decoding and improved performance for longer codes.

Findings

Near-ML performance for short RM codes.

Improved decoding performance for longer RM codes.

Permutation-based decoding enhances error-rate performance.

Abstract

Recursive list decoding of Reed-Muller (RM) codes, with moderate list size, is known to approach maximum-likelihood (ML) performance of short length $(\leq 256)$ RM codes. Recursive decoding employs the Plotkin construction to split the original code into two shorter RM codes with different rates. In contrast to the standard approach which decodes the lower-rate code first, the method in this paper decodes the higher-rate code first. This modification enables an efficient permutation-based decoding technique, with permutations being selected on the fly from the automorphism group of the code using soft information from a channel. Simulation results show that the error-rate performance of the proposed algorithms, enhanced by a permutation selection technique, is close to that of the automorphism-based recursive decoding algorithm with similar complexity for short RM codes, while our decoders perform better for longer RM codes. In particular, it is demonstrated that the proposed algorithms achieve near-ML performance for short RM codes and for RM codes of length $2^m$ and order $m - 3$ with reasonable complexity.