Decoding Reed-Muller and Polar Codes by Successive Factor Graph Permutations

TL;DR

This paper introduces a successive permutation scheme for decoding Reed-Muller and polar codes, improving error correction performance without increasing memory use by dynamically selecting permutations during decoding.

Contribution

The paper proposes a novel successive permutation (SP) decoding scheme that enhances RM and polar codes' error correction by on-the-fly permutation selection, reducing complexity compared to exhaustive permutation methods.

Findings

Up to 0.5 dB improvement for Reed-Muller codes at FER 10^-4.

Up to 0.1 dB improvement for polar codes at FER 10^-4.

SP scheme maintains decoder memory requirements while boosting performance.

Abstract

Reed-Muller (RM) and polar codes are a class of capacity-achieving channel coding schemes with the same factor graph representation. Low-complexity decoding algorithms fall short in providing a good error-correction performance for RM and polar codes. Using the symmetric group of RM and polar codes, the specific decoding algorithm can be carried out on multiple permutations of the factor graph to boost the error-correction performance. However, this approach results in high decoding complexity. In this paper, we first derive the total number of factor graph permutations on which the decoding can be performed. We further propose a successive permutation (SP) scheme which finds the permutations on the fly, thus the decoding always progresses on a single factor graph permutation. We show that SP can be used to improve the error-correction performance of RM and polar codes under successive-cancellation (SC) and SC list (SCL) decoding, while keeping the memory requirements of the decoders unaltered. Our results for RM and polar codes of length $128$ and rate $0.5$ show that when SP is used and at a target frame error rate of $10^{-4}$, up to $0.5$ dB and $0.1$ dB improvement can be achieved for RM and polar codes respectively.