For polar codes, the statistical breakdown of belief propagation (BP) decoding errors is firstly proposed by Sun et al. The errors are classified into three categories: unconverged errors, false converged errors and oscillation errors, which are corrected by three different post-processing methods. As for the BP List (BPL) decoding, we discover that one type of error can be transformed to another type by different factor graphs in the failed BP decoding. Besides, false converged error is usually easier to be detected and modified. Therefore, we only target false converged error to detect and correct. In this work, we propose a two-level detection rule for false converged errors and its corresponding post-processing algorithm to modify these errors in order to improve the performance of the BPL decoding. Numerical results show that the error-correction performance of the proposed decoder is more than 0.85dB better than that of the permuted BPL (PBPL) decoder with slight extra computation complexity at the frame error rate (FER) of 10-5 for 5G (1024,512) polar code.

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 emerging Massive Communication (MC) highlights the need for efficient short-length error-correction coding schemes with effective decoders. Classical Reed-Muller (RM) codes combined with the recently developed Recursive Projection-Aggregation (RPA) decoder present a promising solution, as the RPA decoder demonstrates near Maximum Likelihood (ML) performance and supports highly parallel implementation. To address the speed and flexibility requirements of Cloud Radio Access Networks (C-RANs) across various MC applications, this paper proposes a fast and scalable RPA decoder on Graphics Processing Units (GPUs). By leveraging a thread-per-projection mapping strategy, we develop an optimized thread block architecture for the RPA decoding of second-order RM codes, which can be easily extended to construct a multi-dimensional block array for decoding higher-order RM codes. Additionally, we introduce a stationary projection pruning technique that seamlessly adapts the RPA decoder kernel to simplified variants, facilitating flexible trade-offs between error-correction performance and implementation complexity. Experimental results show that the pruned RPA decoder kernel on the NVIDIA A100 GPU achieves throughputs of 1.69 Gbps and 1.33 Gbps for the RM (6, 2) and RM (7, 2) codes, respectively, delivering speedups of 2.95× and 3.69× compared to a state-of-the-art software-based Successive Cancellation List (SCL) decoder.

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.

The recently developed projection-aggregation (PA) decoder for Reed-Muller (RM) codes delivers superior performance with appealing parallelizability. However, its recursive nature incurs high complexity and intricate scheduling due to the deployment of numerous fast Hadamard transform (FHT) decoders, hindering efficient implementations. In this paper, we propose a graphics processing unit (GPU)-based PA decoder that employs a warp-per-FHT mapping strategy, enabling efficient parallelization of FHT decoding with branch divergence-free thread warp execution. Additionally, we introduce a straightforward data addressing scheme that provably avoids bank conflicts during shared memory access. Experimental results demonstrate that our GPU-based PA decoder achieves throughput speedups of up to $10.64 \times$ compared to related works.

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%.