In this paper, we propose a policy-guided Monte Carlo Tree Search (MCTS) decoder that achieves near maximum-likelihood decoding (MLD) performance for short block codes. The MCTS decoder searches for test error patterns (TEPs) in the received information bits and obtains codeword candidates through re-encoding. The TEP search is executed on a tree structure, guided by a neural network policy trained via MCTS-based learning. The trained policy guides the decoder to find the correct TEPs with minimal steps from the root node (all-zero TEP). The decoder outputs the codeword with maximum likelihood when the early stopping criterion is satisfied. The proposed method requires no Gaussian elimination (GE) compared to ordered statistics decoding (OSD) and can reduce search complexity by 95\% compared to non-GE OSD. It achieves lower decoding latency than both OSD and non-GE OSD at high SNRs.

Traditional wireless communications rely solely on bit-level channel coding for error correction, without exploiting the inherent linguistic structure of the data source. This paper proposes a large language model (LLM) Viterbi decoder that integrates LLM priors into the Viterbi decoding for text transmission over AWGN channels. The proposed decoder maintains multiple candidate paths during the Viterbi decoding and periodically evaluates path reliabilities using a fine-tuned Byte-level T5 (ByT5) language model. By combining channel reliability metrics with semantic probability from the LLM, it outputs the path that maximizes the joint likelihood of channel observations and linguistic coherence. Simulations show that our decoder achieves significant performance gains over conventional Viterbi decoding in terms of both block error rate (BLER) and semantic similarity. For convolutional codes with constraint length 3, it achieves approximately 1.5 dB more coding gain in BLER, with over 50% improvements in semantic similarity. The framework can extend to other structured data sources beyond text.

This paper presents the analysis and a proof-of-concept design of the low-complexity early termination hybrid automatic repeat request (ET-HARQ) for ultra-reliable low-latency communication (URLLC). In ET-HARQ, unfit packets are flagged for retransmission prior to decoding, a process that significantly diminishes decoding complexity and facilitates swift HARQ reporting. This characteristic makes ET-HARQ well-suited for URLLC applications. We analyze the impact of ET-HARQ on the packet error rate (PER), throughput, and complexity performance in the finite block length regime, taking into consideration cyclic redundancy check (CRC) limitations. Numerical results indicate that ET-HARQ significantly reduces the decoding complexity and improves throughput with little to no loss in the PER. In addition, ET-HARQ demonstrates resilience even with a short CRC, whereas the imperfections of a short CRC significantly impact PER reliability in standard HARQ. To validate our analysis, we also design a practical early termination mechanism involving belief propagation and neural network (BP-NN) to predict the decodability of the received packet. Testing with BCH and CRC-polar codes shows that it can reach up to a 70 % ∼ 80 % prediction accuracy with packet lengths less than 128 bits encoded with high-density linear block codes. Simulation shows that the BP-NN-based ET-HARQ has significantly lower complexity at this accuracy level than the standard HARQ with similar reliability.

Reconfigurable intelligent surface (RIS) has emerged as a promising technology for future 6G wireless communications. However, the passive nature of RIS and the high-dimensional cascaded channels pose significant challenges for channel estimation (CE), particularly in practical scenarios where decomposition dictionaries cannot be predefined. This paper proposes a novel three-stage joint CE and positioning (JCEP) framework for RIS-assisted communication systems. It first performs the initial CE based on a predefined row dictionary that exploits the structural properties of cascaded channels, and then conducts positioning based on the initial CE results. Finally, it refines the CE results by incorporating the positioning output to construct customized column dictionaries. The framework employs a unitary approximate message passing sparse Bayesian learning (UAMP-SBL) based channel estimator that adapts to both initial and CE refinement stages. For positioning, we design a graph attention network (GAT) to achieve robust positioning performance in dynamic environments. Furthermore, in the CE refinement, we introduce a location-aware dictionary design that leverages position priors to reduce computational overhead. Additionally, we employ meta-learning to enable rapid adaptation to new environments. Extensive simulations show that our framework achieves superior performance in CE and positioning accuracy with low complexity.

Non-binary convolutional codes (NBCCs) offer significant performance advantages in modern communication systems, but their optimal decoding using classical maximum a posteriori probability (MAP) algorithms is computationally intensive. This paper proposes a low-complexity linear MAP (LMAP) decoding method for rate-1/2 NBCCs. By representing the MAP forward and backward decoding processes as shift register structures operating on probability mass functions (PMFs) of signal estimates, the method supports single direction (forward and backward) SISO decoding, and can achieve the optimal bidirectional MAP decoding performance. The decoder structure is determined offline, and the decoding process only involves simple shift register operations, finite field convolutions, and permutations. Simulation results demonstrate that the proposed LMAP decoder achieves identical error performance to the conventional BCJR MAP decoder, while significantly reducing decoding latency and hardware complexity.

In this paper, we propose a linear representation of BCJR maximum a posteriori probability (MAP) decoding of a rate$1 / 2$convolutional code (CC), referred to as the linear MAP decoding (LMAP). We discover that the MAP forward and backward decoding can be implemented by the corresponding dual soft input and soft output (SISO) encoders using shift registers. The bidrectional MAP decoding output can be obtained by combining the contents of respective forward and backward dual encoders. Represented using simple shift-registers, LMAP decoder maps naturally to hardware registers and thus can be easily implemented. Simulation results demonstrate that the LMAP decoding achieves the same performance as the BCJR MAP decoding, but has a significantly reduced decoding delay. For the block length 64, the CC of the memory length 14 with LMAP decoding surpasses the random coding union (RCU) bound by approximately 0.5 dB at a BLER of$10^{-3}$, and closely approaches both the normal approximation (NA) and meta-converse (MC) bounds.

This paper provides the first analytical characterization of the achievable guesswork complexity of ordered statistics decoding (OSD) in binary AWGN channels. This complexity is defined as the number of test error patterns (TEPs) processed by OSD immediately upon finding the correct codeword estimate. We show that for an order-$k$OSD of a$(n, k)$code, the average achievable guesswork complexity is tightly approximated by$e^{-k p_{e}} I_{0}\left(2 k \sqrt{p_{e}}\right)$, where$I_{0}$is the modified Bessel function and$p_{e}$is determined by the code rate and SNR. Furthermore, for an order-$m$OSD$(m

This paper presents a novel auto-encoder based end-to-end channel encoding and decoding. It integrates deep reinforcement learning (DRL) and graph neural networks (GNN) in code design by modeling the generation of code parity-check matrices as a Markov Decision Process (MDP), to optimize key coding performance metrics such as error-rates and code algebraic properties. An edge-weighted GNN (EW-GNN) decoder is proposed, which operates on the Tanner graph with an iterative message-passing structure. Once trained on a single linear block code, the EW-GNN decoder can be directly used to decode other linear block codes of different code lengths and code rates. An iterative joint training of the DRL-based code designer and the EW-GNN decoder is performed to optimize the end-end encoding and decoding process. Simulation results show the proposed auto-encoder significantly surpasses several traditional coding schemes at short block lengths, including low-density parity-check (LDPC) codes with the belief propagation (BP) decoding and the maximum-likelihood decoding (MLD), and BCH with BP decoding, offering superior error-correction capabilities while maintaining low decoding complexity.

Sparse vector coding (SVC) is a promising coding technique to achieve high transmission reliability and low latency for short packet communications. However, for SVC with conventional combination-based sparse mapping, a small increase of transmitted bits may lead to excessively long sparse vectors, resulting in unsatisfactory transmission performance when coding efficiency is high. In this paper, we propose a generalized index redefinition (IR)-based SVC (GIR-SVC) to significantly enhance the efficiency of SVC. The IR mechanism enables multiple index bit streams to share position resources in SVC, with the help of constellation labels. GIR-SVC constructs the sparse vector using a hybrid IR mechanism that integrates the unlabeled IR and the pairwise-grouping-based labeled IR, which allows efficient mapping and de-mapping of index bits without requiring index tables. Consequently, the proposed GIR-SVC can be efficiently decoded without the index table using sparse recovery algorithms. Theoretical analysis is conducted to validate the block error rate (BLER) performance of GIR-SVC. Simulations show that GIR-SVC can significantly reduce the decoding delay compared to existing approaches, while maintaining the high transmission reliability.

This paper investigates guesswork over ordered statistics and formulates the achievable guesswork complexity of ordered statistics decoding (OSD) in binary additive white Gaussian noise (AWGN) channels. The achievable guesswork complexity is defined as the number of test error patterns (TEPs) processed by OSD immediately upon finding the correct codeword estimate. The paper first develops a new upper bound for guesswork over independent sequences by partitioning them into Hamming shells and applying Hölder’s inequality. This upper bound is then extended to ordered statistics, by constructing the conditionally independent sequences within the ordered statistics sequences. Next, we apply these bounds to characterize the statistical moments of the OSD guesswork complexity. We show that the achievable guesswork complexity of OSD at maximum decoding order can be accurately approximated by the modified Bessel function, which increases exponentially with code dimension. We also identify a guesswork complexity saturation threshold, where increasing the OSD decoding order beyond this threshold improves error performance without further raising the achievable guesswork complexity. Finally, the paper presents insights on applying these findings to enhance the design of OSD decoders.

In this paper, we propose an efficient decoding algorithm for short low-density parity check (LDPC) codes by carefully combining the belief propagation (BP) decoding and ordered statistics decoding (OSD) algorithms. Specifically, a modified BP (mBP) algorithm is applied for a certain number of iterations prior to OSD to enhance the reliability of the received message, where an offset parameter is utilized in mBP to control the weight of the extrinsic information in message passing. By carefully selecting the offset parameter and the number of mBP iterations, the number of errors in the most reliable positions (MRPs) in OSD can be reduced by mBP, thereby significantly improving the overall decoding performance of error rate and complexity. Simulation results show that the proposed algorithm can approach the maximum-likelihood decoding (MLD) for short LDPC codes with only a slight increase in complexity compared to BP and a significant decrease compared to OSD. Specifically, the order- (m -1) decoding of the proposed algorithm can achieve the performance of the order-m OSD.

In this work, we propose a novel decoding algorithm for short block codes based on an edge-weighted graph neural network (EW-GNN). The EW-GNN decoder operates on the Tanner graph with an iterative message-passing structure, which algorithmically aligns with the conventional belief propagation (BP) decoding method. In each iteration, the “weight” on the message passed along each edge is obtained from a fully connected neural network that has the reliability information from nodes/edges as its input. Compared to existing deep-learning-based decoding schemes, the EW-GNN decoder is characterised by its scalability, meaning that 1) the number of trainable parameters is independent of the codeword length, and 2) an EW-GNN decoder trained with shorter/simple codes can be directly used for longer/sophisticated codes of different code rates. Furthermore, simulation results show that the EW-GNN decoder outperforms the BP and deep-learning-based BP methods from the literature in terms of the decoding error rate.

In this paper, we design the joint decoding (JD) of non-orthogonal multiple access (NOMA) systems employing short block length codes. We first proposed a low-complexity soft-output ordered-statistics decoding (LC-SOSD) based on a decoding stopping condition, derived from approximations of the a-posterior probabilities of codeword estimates. Simulation results show that LC-SOSD has the similar mutual information transform property to the original SOSD with a significantly reduced complexity. Then, based on the analysis, an efficient JD receiver which combines the parallel interference cancellation (PIC) and the proposed LC-SOSD is developed for NOMA systems. Two novel techniques, namely decoding switch (DS) and decoding combiner (DC), are introduced to accelerate the convergence speed. Simulation results show that the proposed receiver can achieve a lower bit-error rate (BER) compared to the successive interference cancellation (SIC) decoding over the additive-white-Gaussian-noise (AWGN) and fading channel, with a lower complexity in terms of the number of decoding iterations.

In this paper, we propose a new linear-equation ordered-statistics decoding (LE-OSD). Unlike the OSD, LE-OSD uses high reliable parity bits rather than information bits to recover codeword estimates, which is equivalent to solving a system of linear equations (SLE). Only test error patterns (TEPs) that create feasible SLEs, referred to as the valid TEPs, are used to obtain codeword estimates. We introduce several constraints on the Hamming weight of TEPs to limit the overall decoding complexity. Furthermore, we analyze the block error rate (BLER) and the computational complexity of the proposed approach. It is shown that LE-OSD has a similar performance to OSD in terms of BLER, which can asymptotically approach Maximum-likelihood (ML) performance with proper parameter selections. Simulation results demonstrate that the LE-OSD has a significantly reduced complexity compared to OSD, especially for low-rate codes, that usually require high decoding order in OSD. Nevertheless, the complexity reduction can also be observed for high-rate codes. In addition, we further improve LE-OSD by applying the decoding stopping condition and the TEP discarding condition. As shown by simulations, the improved LE-OSD has a considerably reduced complexity while maintaining the BLER performance, compared to the latest OSD approaches from literature.

This paper revisits the ordered statistics decoding (OSD). It provides a comprehensive analysis of the OSD algorithm by characterizing the statistical properties, evolution and the distribution of the Hamming distance and weighted Hamming distance from codeword estimates to the received sequence in the reprocessing stages of the OSD algorithm. We prove that the Hamming distance and weighted Hamming distance distributions can be characterized as mixture models capturing the decoding error probability and code weight enumerator. Simulation and numerical results show that our proposed statistical approaches can accurately describe the distance distributions. Based on these distributions and with the aim to reduce the decoding complexity, several techniques, including stopping rules and discarding rules, are proposed, and their decoding error performance and complexity are accordingly analyzed. Simulation results for decoding various eBCH codes demonstrate that the proposed techniques can significantly reduce the decoding complexity with a negligible loss in the decoding error performance.

In this paper, we propose an efficient reliability based segmentation-discarding decoding (SDD) algorithm for short block-length codes. A novel segmentation- discarding technique is proposed along with the stopping rule to significantly reduce the decoding complexity without a significant performance degradation compared to ordered statistics decoding (OSD). In the proposed decoder, the list of test error patterns (TEPs) is divided into several segments according to carefully selected boundaries and every segment is checked separately during the reprocessing stage. Decoding is performed under the constraint of the discarding rule and stopping rule. Simulations results for different codes show that our proposed algorithm can significantly reduce the decoding complexity compared to the existing OSD algorithms in literature.

In this paper, we derive the distribution of the Hamming distance at 0-reprocessing of the ordered statistics decoding (OSD). With the assumption of decoding a random linear block code, we first find the distribution of the number of errors in any partition of the ordered channel output sequence. Then the distribution of the Hamming distance after 0-reprocessing is derived by a mixture model of two random variables. Based on the proposed statistical approach, we outline the design of high-efficiency OSD algorithm. Simulation and numerical results show that our proposed statistical approaches accurately describe the Hamming distance distributions in OSD decoding process.