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.
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.
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%.
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.
Projected requirements for 6G ultra-reliable low-latency communications (URLLC) scenario require extremely low latency and efficient error correction akin to maximum likelihood (ML) performance. The quest for codes and decoding methods that balance low latency, energy efficiency, and error correction ability is at its height. Polar codes known for their capacity achievement in symmetric memoryless channels have shown promise in 5G applications. Advancing further existing decoding techniques like successive cancellation list (SCL) algorithm offer a path towards ML performance, whilst at the expense of having increased computational complexity. The introduction of automorphism ensemble (AE) decoding provided a novel approach for parallel decoding, particularly beneficial for codes with rich automorphism group like Reed-Muller (RM) codes. This work introduces a novel modification of AE decoding termed reduced-delay AE (RD-AE) decoder. RD-AE leverages a tailored set of automorphisms with the primary aim at diminishing decoding delay but also offering the flexibility to prioritize either delay reduction or error correction performance.
The anticipated demands of 6G ultra-reliable low-latency communications (URLLC) call for near-instantaneous data transfers and error-correction efficiency rivaling maximum-likelihood (ML) decoding. Achieving an effective compromise between latency, power consumption, and decoding accuracy is a pivotal challenge in contemporary coding theory. Polar codes have distinguished themselves by achieving capacity on symmetric memoryless channels. To narrow the gap to ML performance, refined techniques like successive-cancellation list (SCL) decoding have been developed, albeit with a rise in computational burden. More recently, automorphism ensemble (AE) decoding has unlocked parallelization benefits for codes with rich automorphism groups, such as Reed-Muller codes. This paper discusses the reduced-delay AE (RD-AE) decoder, which harnesses a carefully chosen subset of automorphisms to drastically cut decoding latency while allowing a flexible trade-off between speed and error-correction performance. This article is an extended version of research previously introduced at a conference. We discuss key aspects and nuances of the algorithm and its implementation, including: (1) algorithms for selecting automorphism sets to meet specific performance targets, (2) a detailed procedure for combining bits from different permutations, (3) code construction methodologies for RD-AE decoding, and (4) implementation considerations. Furthermore, we explore various scenarios and applications of the proposed algorithm and present simulation results demonstrating significant latency reductions while maintaining error-correction performance.
Automorphism Ensemble (AE) decoding has recently drawn attention as a possible alternative to list decoding of polar codes. In this letter, we investigate the distribution of Partially-Symmetric Reed-Muller (PS-RM) codes, a family of polar codes yielding good performances under AE decoding. We prove the existence of these codes for almost all code dimensions for code lengths N ≤ 256. Moreover, we analyze the absorption group of this family of codes under SC decoding, proving that valuable permutations in AE decoding always exist. Finally, we experimentally show that PS-RM codes can outperform state-of-the-art polar-code-construction algorithms in terms of error-correction performance for short code lengths, while reducing decoding latency.