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Entropy

Low-Complexity Automorphism Ensemble Decoding of Reed-Muller Codes Using Path Pruning

Kairui Tian, Rongke Liu, Zheng Lu

Published Jul 28, 2025• 0 citations
Publisher

Abstract

The newly developed automorphism ensemble decoder (AED) leverages the rich automorphisms of Reed-Muller (RM) codes to achieve near maximum likelihood (ML) performance at short code lengths. However, the performance gain of AED comes at the cost of high complexity, as the ensemble size required for near ML decoding grows exponentially with the code length. In this work, we address this complexity issue by focusing on the factor graph permutation group (FGPG), a subgroup of the full automorphism group of RM codes, to generate permutations for AED. We propose a uniform partitioning of FGPG based on the affine bijection permutation matrices of automorphisms, where each subgroup of FGPG exhibits permutation invariance (PI) in a Plotkin construction-based information set partitioning for RM codes. Furthermore, from the perspective of polar codes, we exploit the PI property to prove a subcode estimate convergence (SEC) phenomenon in the AED that utilizes successive cancellation (SC) or SC list (SCL) constituent decoders. Observing that strong SEC correlates with low noise levels, where the full decoding capacity of AED is often unnecessary, we perform path pruning to reduce the decoding complexity without compromising the performance. Our proposed SEC-aided path pruning allows only a subset of constituent decoders to continue decoding when the intensity of SEC exceeds a preset threshold during decoding. Numerical results demonstrate that, for the FGPG-based AED of various short RM codes, the proposed SEC-aided path pruning technique incurs negligible performance degradation, while achieving a complexity reduction of up to 67.6%.

DOI
10.3390/e27080808

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Decoding of Reed-Muller Codes Starting With a Higher-Rate Constituent Code.

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•Nov 28, 2021•0 citations
The newly developed automorphism ensemble decoder (AED) leverages the rich automorphisms of Reed-Muller (RM) codes to achieve near maximum likelihood (ML) performance at short code lengths. However, the performance gain of AED comes at the cost of high complexity, as the ensemble size required for near ML decoding grows exponentially with the code length. In this work, we address this complexity issue by focusing on the factor graph permutation group (FGPG), a subgroup of the full automorphism group of RM codes, to generate permutations for AED. We propose a uniform partitioning of FGPG based on the affine bijection permutation matrices of automorphisms, where each subgroup of FGPG exhibits permutation invariance (PI) in a Plotkin construction-based information set partitioning for RM codes. Furthermore, from the perspective of polar codes, we exploit the PI property to prove a subcode estimate convergence (SEC) phenomenon in the AED that utilizes successive cancellation (SC) or SC list (SCL) constituent decoders. Observing that strong SEC correlates with low noise levels, where the full decoding capacity of AED is often unnecessary, we perform path pruning to reduce the decoding complexity without compromising the performance. Our proposed SEC-aided path pruning allows only a subset of constituent decoders to continue decoding when the intensity of SEC exceeds a preset threshold during decoding. Numerical results demonstrate that, for the FGPG-based AED of various short RM codes, the proposed SEC-aided path pruning technique incurs negligible performance degradation, while achieving a complexity reduction of up to 67.6%.
By exploiting the rich automorphisms of Reed-Muller (RM) codes, the recently developed automorphism ensemble (AE) successive cancellation (SC) decoder achieves a near-maximum-likelihood (ML) performance for short block lengths. However, the appealing performance of AE-SC decoding arises from the diversity gain that requires a list of SC decoding attempts, which results in a high decoding complexity. To address this issue, this paper proposes a novel quasi-optimal path convergence (QOPC)-aided early termination (ET) technique for AE-SC decoding. This technique detects strong convergence between the partial path metrics (PPMs) of SC constituent decoders to reliably identify the optimal decoding path at runtime. When the QOPC-based ET criterion is satisfied during the AE-SC decoding, only the identified path is allowed to proceed for a complete codeword estimate, while the remaining paths are terminated early. The numerical results demonstrated that for medium-to-high-rate RM codes in the short-length regime, the proposed QOPC-aided ET method incurred negligible performance loss when applied to fully parallel AE-SC decoding. Meanwhile, it achieved a complexity reduction that ranged from 35.9% to 47.4% at a target block error rate (BLER) of 10-3, where it consistently outperformed a state-of-the-art path metric threshold (PMT)-aided ET method. Additionally, under a partially parallel framework of AE-SC decoding, the proposed QOPC-aided ET method achieved a greater complexity reduction that ranged from 81.3% to 86.7% at a low BLER that approached 10-5 while maintaining a near-ML decoding performance.
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.
Automorphism ensemble decoding (AED) is a highly parallel approach that enables decoding of polar and Reed-Muller (RM) codes with automorphisms, offering a practical solution with near-maximum likelihood (ML) performance and manageable computational complexity. To meet the growing demands for high throughput and low latency in cloud and virtual random access networks, this paper presents a graphics processing unit (GPU)-based AED architecture for polar and RM codes, utilizing low-complexity successive cancellation (SC) and small list SC (SCL) decoders as the constituent of AED. The proposed architecture exploits the inherent parallelism of AED to optimize decoding tasks on the GPU, significantly enhancing throughput by efficiently harnessing the extensive parallel processing capabilities of the GPU. Additionally, improved thread mapping and data management techniques markedly reduce latency for automorphism ensemble SC (Aut-SC) decoding, while a low-latency sorting mechanism further accelerates automorphism ensemble SCL (Aut-SCL) decoding. Experimental results on an NVIDIA RTX 4090 demonstrate that the proposed Aut-SC decoder, with an ensemble size of 8, achieves a throughput exceeding 17 Gbps. Compared to the state-of-the-art softwarebased SCL decoders, the proposed GPU-based Aut-SC and Aut-SCL architectures outperform existing solutions by factors of up to 28× and 10×, respectively, in normalized throughput while maintaining the same or even superior error correction performance.
Recursive list decoding of Reed-Muller (RM) codes, with moderate list size, is known to approach maximum-likelihood (ML) performance of short length(≤256)(\leq 256)(≤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 length2m2^{m}2mand orderm−3m-3m−3with reasonable complexity.
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.
List versions of recursive decoding are known to approach maximum-likelihood (ML) performance for the short length Reed-Muller (RM) codes. The recursive decoder employs the Plotkin construction to split the original code into two shorter RM codes, with a lower-rate RM code being decoded first. In this paper, we consider non-iterative soft-input decoders for RM codes that, unlike recursive decoding, start decoding with a higher-rate constituent code. Although the error-rate performance of these algorithms is limited, it can be significantly improved by applying permutations from the automorphism group of the code, with permutations being selected on the fly based on 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 length (≤256)(\leq 256)(≤256) 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 RM codes of length 2m2^m2m and order m−3m - 3m−3 with reasonable complexity.