Mohsen Moradi, Amir Mozammel, Kangjian Qin, Erdal Arikan
Performance and complexity of sequential decoding of polarization-adjusted convolutional (PAC) codes is studied. In particular, a performance and computational complexity comparison of PAC codes with 5G polar codes and convolutional codes is given. A method for bounding the complexity of sequential decoding of PAC codes is proposed.
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Simulation results illustrating the performance and complexity of the sequential successive cancellation decoding algorithm are presented for the case of polar subcodes with Arikan and large kernels, as well as for extended BCH\ codes. Performance comparison with Arikan PAC and LDPC codes is provided. Furthermore, complete description of the decoding algorithm is presented.
A reduced complexity sequential decoding algorithm for polar (sub)codes is described. The proposed approach relies on a decomposition of the polar (sub)code being decoded into a number of outer codes, and on-demand construction of codewords of these codes in the descending order of their probability. Construction of such codewords is implemented by fast decoding algorithms, which are available for many codes arising in the decomposition of polar codes. Further complexity reduction is achieved by taking hard decisions of the intermediate LLRs, and avoiding decoding of some outer codes. Data str…
Sequential Decoding of Convolutional Codes for Synchronization Errors
本文提出一种适用于易发生插入、删除和替换误差信道的卷积码序列解码器，通过引入漂移状态扩展码树，并改进Fano度量，降低解码复杂度，但牺牲少量误码率。
A new score function is proposed for stack decoding of polar codes, which enables one to accurately compare paths of different lengths. The proposed score function includes bias, which reflects the average behaviour of the correct path. This enables significant complexity reduction with respect to the original stack algorithm at the expense of a negligible performance loss.
Despite the extreme error-correction performance, the amount of computation of sequential decoding of the polarization-adjusted convolutional (PAC) codes is random. In sequential decoding of convolutional codes, the computational cutoff rate denotes the region between rates whose average computational complexity of decoding is finite and those which is infinite. In this paper, by benefiting from the polarization and guessing techniques, we prove that the computational cutoff rate in sequential decoding of pre-transformed polar codes polarizes. The polarization of the computational cutoff rate …
Simplified Successive Cancellation List Decoding of PAC Codes
针对PAC码的SCL译码延迟问题，提出通过识别译码树特殊节点并在中间阶段处理的SSCL算法，在保持近似相同性能的同时降低译码延迟。
In this paper, the asymptotic performance of the lattice sequential decoder for LAttice Space-Time (LAST) coded MIMO channel is analyzed. We determine the rates achievable by lattice coding and sequential decoding applied to such a channel. The diversity-multiplexing tradeoff (DMT) under lattice sequential decoding is derived as a function of its parameter---the bias term, which is critical for controlling the amount of computations required at the decoding stage. Achieving low decoding complexity requires increasing the value of the bias term. However, this is done at the expense of losing th…
Sequential Decoding of Multiple Traces Over the Syndrome Trellis for Synchronization Errors
针对高码率卷积码同步错误联合译码复杂度过高的问题，提出校正子网格上多序列联合序列译码方法，短块长下性能优于分离式BCJR译码且复杂度更低，还扩展出双向模式以降低译码超时概率。
Polar coding gives rise to the first explicit family of codes that provably achieve capacity with efficient encoding and decoding for a wide range of channels. However, its performance at short block lengths is far from optimal. Arikan has recently presented a new polar coding scheme, which he called polarization-adjusted convolutional (PAC) codes. Such PAC codes provide dramatic improvement in performance as compared to both standard successive-cancellation decoding as well as CRC-aided list decoding. Arikan's PAC codes are based primarily upon the following ideas: replacing CRC precoding wit…
The optimal diversity-multiplexing-delay tradeoff for the multi-input multi-output (MIMO) automatic repeat request (ARQ) channel can be achieved using incremental redundancy lattice space-time codes coupled with a list decoder for joint error detection and correction. Such a decoder is based on the minimum mean-square error lattice decoding principle which is implemented using sphere decoding algorithms. However, sphere decoders suffer from high computational complexity for low-to-moderate signal-to-noise ratios, especially for large signal dimensions. In this paper, we would like to construct…
In this paper, the performance limits and the computational complexity of the lattice sequential decoder are analyzed for the unconstrained additive white Gaussian noise channel. The performance analysis available in the literature for such a channel has been studied only under the use of the minimum Euclidean distance decoder that is commonly referred to as the lattice decoder. Lattice decoders based on solutions to the NP-hard closest vector problem are very complex to implement, and the search for low complexity receivers for the detection of lattice codes is considered a challenging proble…
A sequential updating scheme (SUS) for belief propagation (BP) decoding of LDPC codes over Galois fields, $GF(q)$, and correlated Markov sources is proposed, and compared with the standard parallel updating scheme (PUS). A thorough experimental study of various transmission settings indicates that the convergence rate, in iterations, of the BP algorithm (and subsequently its complexity) for the SUS is about one half of that for the PUS, independent of the finite field size $q$. Moreover, this 1/2 factor appears regardless of the correlations of the source and the channel's noise model, while t…
Sequential Automorphism Ensemble Decoding with Early Stopping
本文提出了一种低复杂度的自同构集合解码器（AED）方法，利用逐次取消（SC）作为构成解码器，通过顺序激活子解码器并在预优化参数下终止解码过程，以减少平均解码复杂度并满足块错误率（BLER）约束。
This note is a written and extended version of the Shannon Lecture I gave at 2019 International Symposium on Information Theory. It gives an account of the original ideas that motivated the development of polar coding and discusses some new ideas for exploiting channel polarization more effectively in order to improve the performance of polar codes.
PAC Codes: Sequential Decoding vs List Decoding
本文研究PAC码的译码方法，将列表译码和顺序译码适配用于PAC码，并提出自适应度量、树搜索约束和搜索策略以降低顺序译码复杂度，同时给出中间LLR的高效计算方法，最后比较三种译码算法的性能、复杂度与资源需求。
Reinforcement Learning for Sequential Decoding of Generalized LDPC Codes
本文提出用强化学习优化广义LDPC码的序贯解码调度策略，以最小化BP解码复杂度，显著优于标准泛洪和随机调度方法。
A latest coding scheme named polarization-adjusted convolutional (PAC) codes is shown to approach the dispersion bound for the code (128,64) under list decoding. However, to achieve the near-bound performance, the list size of list decoding needs to be excessively large, which leads to insufferable latency. In this paper, to improve the speed of list decoding, fast list decoders for PAC codes are proposed. We define four types of constituent nodes and provide fast list decoding algorithms for each of them. Simulation results present that fast list decoding with three types of constituent nodes…
his study presents a novel technique to estimate the computational complexity of sequential decoding using the Berry-Esseen theorem. Unlike the theoretical bounds determined by the conventional central limit theorem argument, which often holds only for sufficiently large codeword length, the new bound obtained from the Berry-Esseen theorem is valid for any blocklength. The accuracy of the new bound is then examined for two sequential decoding algorithms, an ordering-free variant of the generalized Dijkstra's algorithm (GDA)(or simplified GDA) and the maximum-likelihood sequential decoding algo…
Decoding Golay Codes and their Related Lattices: A PAC Code Perspective
从PAC码视角提出解码Golay码的方法，实现高效并行列表解码，适用于解码Golay码及相关晶格。
Performance Evaluation of PAC Decoding with Deep Neural Networks
本文针对传统 PAC 解码延迟高的问题，提出了 MLP、CNN 和 RNN 三种深度神经网络解码器，仿真表明在参数量相近时 MLP 解码器纠错性能最佳。
Motivated by video coding applications, the problem of sequential coding of correlated sources with encoding and/or decoding frame-delays is studied. The fundamental tradeoffs between individual frame rates, individual frame distortions, and encoding/decoding frame-delays are derived in terms of a single-letter information-theoretic characterization of the rate-distortion region for general inter-frame source correlations and certain types of potentially frame specific and coupled single-letter fidelity criteria. The sum-rate-distortion region is characterized in terms of generalized directed …
A common problem on sequential-type decoding is that at the signal-to-noise ratio (SNR) below the one corresponding to the cutoff rate, the average decoding complexity per information bit and the required stack size grow rapidly with the information length. In order to alleviate the problem in the maximum-likelihood sequential decoding algorithm (MLSDA), we propose to directly eliminate the top path whose end node is $Δ$-trellis-level prior to the farthest one among all nodes that have been expanded thus far by the sequential search. Following random coding argument, we analyze the early-elimi…
A Novel SCL Bit-Flipping Decoding Of Polarization-Adjusted Convolutional (PAC) Codes
针对 PAC 码短码长译码，本文提出结合 SCL 与比特翻转后处理的 PAC-SCLF，在最多翻转 5 比特时以 L=32 达到接近 PAC-SCL 的 L=128 性能，并提升 0.3 dB。
Reverse PAC Codes: Look-ahead List Decoding
本文研究Reverse PAC Codes以克服PAC码减少最小重量码字的局限，提出一种复杂度同级的前瞻列表译码，显著降低错误系数并改善高码率下的块错误率。
Convolutional precoding in polarization-adjusted convolutional (PAC) codes can reduce the number of minimum weight codewords (a.k.a error coefficient) of polar codes. This can result in improving the error correction performance of (near) maximum likelihood (ML) decoders such as sequential decoders and sphere decoders. However, PAC codes cannot be decoded by sphere decoding. The reason is twofold: 1) Sphere decoding of polar codes is performed from the last bit - due to the lower rectangular shape of the polar transform. Whereas the shape of PAC codes generator matrix is no longer triangular. …
Sequential BP-based Decoding of QLDPC Codes
本研究提出顺序调度改进BP解码方法，提升QLDPC码解码的可靠性和效率。
Polarization-adjusted convolutional (PAC) codes are special concatenated codes in which we employ a one-to-one convolutional transform as a pre-coding step before the polar transform. In this scheme, the polar transform (as a mapper) and the successive cancellation process (as a demapper) present a synthetic vector channel to the convolutional transformation. The numerical results show that this concatenation improves the Hamming distance properties of polar codes. In this work, we implement the parallel list Viterbi algorithm (LVA) and show how the error correction performance moves from the …
Efficient Low-Memory Fast Stack Decoding with Variance Polarization for PAC Codes
本文提出改进的PAC码栈解码算法，通过专用比特节点提升并行化，引入剪枝技术减少路径数量，结合方差极化近似方法降低解码延迟和复杂度，实验显示在PAC(128,64)码中路径数减少70%而不影响纠错性能。
In this paper, we present an optimal metric function on average, which leads to a significantly low decoding computation while maintaining the superiority of the polarization-adjusted convolutional (PAC) codes' error-correction performance. With our proposed metric function, the PAC codes' decoding computation is comparable to the conventional convolutional codes (CC) sequential decoding. Moreover, simulation results show an improvement in the low-rate PAC codes' error-correction performance when using our proposed metric function. We prove that choosing the polarized cutoff rate as the metric…
We consider binary systematic network codes and investigate their capability of decoding a source message either in full or in part. We carry out a probability analysis, derive closed-form expressions for the decoding probability and show that systematic network coding outperforms conventional network coding. We also develop an algorithm based on Gaussian elimination that allows progressive decoding of source packets. Simulation results show that the proposed decoding algorithm can achieve the theoretical optimal performance. Furthermore, we demonstrate that systematic network codes equipped w…
In this paper, we study the power of sequential decoding strategies for several channels with classical input and quantum output. In our sequential decoding strategies, the receiver loops through all candidate messages trying to project the received state onto a `typical' subspace for the candidate message under consideration, stopping if the projection succeeds for a message, which is then declared as the guess of the receiver for the sent message. We show that even such a conceptually simple strategy can be used to achieve rates up to the mutual information for a single sender single receive…
Conflict-Free Color-Clustered Sequential Belief-Propagation Decoding of Quantum LDPC Codes via Reinforcement Learning
本文提出一种基于强化学习的无冲突颜色聚类顺序BP解码方法，通过构建VN冲突图并着色，实现并行更新，在保持性能的同时大幅减少调度决策。
We consider receiver design for coded transmission over linear Gaussian channels. We restrict ourselves to the class of lattice codes and formulate the joint detection and decoding problem as a closest lattice point search (CLPS). Here, a tree search framework for solving the CLPS is adopted. In our framework, the CLPS algorithm decomposes into the preprocessing and tree search stages. The role of the preprocessing stage is to expose the tree structure in a form {\em matched} to the search stage. We argue that the minimum mean square error decision feedback (MMSE-DFE) frontend is instrumental …
The problem of lossless fixed-rate streaming coding of discrete memoryless sources with side information at the decoder is studied. A random time-varying tree-code is used to sequentially bin strings and a Stack Algorithm with a variable bias uses the side information to give a delay-universal coding system for lossless source coding with side information. The scheme is shown to give exponentially decaying probability of error with delay, with exponent equal to Gallager's random coding exponent for sources with side information. The mean of the random variable of computation for the stack deco…
This paper proposes a rate-profile construction method for polarization-adjusted convolutional (PAC) codes of any code length and rate, which is capable of maintaining trade-off between the error-correction performance and decoding complexity of PAC code. The proposed method can improve the error-correction performance of PAC codes while guaranteeing a low mean sequential decoding complexity for signal-to-noise ratio (SNR) values beyond a target SNR value.
Learning-Based List Sequential Belief Propagation Decoding of Quantum LDPC Codes
提出强化学习列表顺序BP译码器，通过列表搜索扩展学习顺序调度，结合路径度量剪枝，在去极化信道下提升QLDPC码译码性能。
Learned Parallel Bit-Flipping Sequential Belief Propagation Decoding of Quantum LDPC Codes
针对量子LDPC码，提出一种结合四元评分与并行比特翻转的序贯BP解码器，通过并行候选延续提升学习型序贯BP的可靠性，并保持低延迟。
Raptor codes have been widely used in many multimedia broadcast/multicast applications. However, our understanding of Raptor codes is still incomplete due to the insufficient amount of theoretical work on the performance analysis of Raptor codes, particularly under maximum-likelihood (ML) decoding, which provides an optimal benchmark on the system performance for the other decoding schemes to compare against. For the first time, this paper provides an upper bound and a lower bound, on the packet error performance of Raptor codes under ML decoding, which is measured by the probability that all …
Sorting operation is one of the main bottlenecks for the successive-cancellation list (SCL) decoding. This paper introduces an improvement to the SCL decoding for polar and pre-transformed polar codes that reduces the number of sorting operations without degrading the code's error-correction performance. In an SCL decoding with an optimum metric function we show that, on average, the correct branch's bit-metric value must be equal to the bit-channel capacity, and on the other hand, the average bit-metric value of a wrong branch can be at most zero. This implies that a wrong path's partial path…
Fountain codes are erasure correcting codes realizing reliable communication systems for the multicast on the Internet. The zigzag decodable fountain (ZDF) code is one of generalization of the Raptor code, i.e, applying shift operation to generate the output packets. The ZDF code is decoded by a two-stage iterative decoding algorithm, which combines the packet-wise peeling algorithm and the bit-wise peeling algorithm. By the bit-wise peeling algorithm and shift operation, ZDF codes outperform Raptor codes under iterative decoding in terms of decoding erasure rate and overhead. However, the bit…
Random linear network coding (RLNC) in theory achieves the max-flow capacity of multicast networks, at the cost of high decoding complexity. To improve the performance-complexity tradeoff, we consider the design of sparse network codes. A generation-based strategy is employed in which source packets are grouped into overlapping subsets called generations. RLNC is performed only amongst packets belonging to the same generation throughout the network so that sparseness can be maintained. In this paper, generation-based network codes with low reception overheads and decoding costs are designed fo…
Since a quantum measurement generally disturbs the state of a quantum system, one might think that it should not be possible for a sender and receiver to communicate reliably when the receiver performs a large number of sequential measurements to determine the message of the sender. We show here that this intuition is not true, by demonstrating that a sequential decoding strategy works well even in the most general "one-shot" regime, where we are given a single instance of a channel and wish to determine the maximal number of bits that can be communicated up to a small failure probability. Thi…
Optimization of Sparse VLSF Codes for Short-Packet Transmission via Saddlepoint Methods
本文提出基于鞍点近似的稀疏变长停止反馈码优化框架，通过高效梯度优化提升解码配置参数性能，改进解码规则并验证其在低计算成本下的近优性。
This paper considers multiplexing two sequences of messages with two different decoding delays over a packet erasure channel. In each time slot, the source constructs a packet based on the current and previous messages and transmits the packet, which may be erased when the packet travels from the source to the destination. The destination must perfectly recover every source message in the first sequence subject to a decoding delay $T_\mathrm{v}$ and every source message in the second sequence subject to a shorter decoding delay $T_\mathrm{u}\le T_\mathrm{v}$. We assume that the channel loss mo…
This paper studies the decoding capabilities of maximum distance profile (MDP) convolutional codes over the erasure channel and compares them with the decoding capabilities of MDS block codes over the same channel. The erasure channel involving large alphabets is an important practical channel model when studying packet transmissions over a network, e.g, the Internet.
PAC codes with Bounded-Complexity Sequential Decoding: Pareto Distribution and Code Design
本文研究了 PAC 码序贯译码的复杂度问题，证明遵循极化截止速率约束可使计算量服从有界 Pareto 分布，并指出基于此设计的 Fano 译码器在 FER 为 $10^{-5}$ 时较 5G 极化码和 LDPC 码获得超 $0.75$ dB 增益。
The polarization-adjusted convolutional (PAC) codes concatenate the polar transform and the convolutional transform to improve the decoding performance of the finite-length polar codes, where the rate-profile is used to construct the PAC codes by setting the positions of frozen bits. However, the optimal rateprofile method of PAC codes is still unknown. In this paper, an optimized rate-profile algorithm of PAC codes is proposed. First, we propose the normalized compression factor (NCF) to quantify the transmission efficiency of useful information, showing that the distribution of useful inform…
Low-Density Parity-Check (LDPC) codes are usually decoded by running an iterative belief-propagation, or message-passing, algorithm over the factor graph of the code. The traditional message-passing schedule consists of updating all the variable nodes in the graph, using the same pre-update information, followed by updating all the check nodes of the graph, again, using the same pre-update information. Recently several studies show that sequential scheduling, in which messages are generated using the latest available information, significantly improves the convergence speed in terms of number …
Speeding up Speculative Decoding via Sequential Approximate Verification
SPRINTER通过训练低复杂度验证器进行顺序近似验证，减少对目标LLM的调用，在保持生成质量的同时进一步降低投机解码的延迟。
Recent work have shown that Reed-Muller (RM) codes achieve the erasure channel capacity. However, this performance is obtained with maximum-likelihood decoding which can be costly for practical applications. In this paper, we propose an encoding/decoding scheme for Reed-Muller codes on the packet erasure channel based on Plotkin construction. We present several improvements over the generic decoding. They allow, for a light cost, to compete with maximum-likelihood decoding performance, especially on high-rate codes, while significantly outperforming it in terms of speed.
In this paper, we are interested in the performance of a variable-length stop-feedback (VLSF) code with $m$ optimal decoding times for the binary-input additive white Gaussian noise channel. We first develop tight approximations on the tail probability of length-$n$ cumulative information density. Building on the work of Yavas \emph{et al.}, for a given information density threshold, we formulate the integer program of minimizing the upper bound on average blocklength over all decoding times subject to the average error probability, minimum gap and integer constraints. Eventually, minimization…
Error control is significant to network coding, since when unchecked, errors greatly deteriorate the throughput gains of network coding and seriously undermine both reliability and security of data. Two families of codes, subspace and rank metric codes, have been used to provide error control for random linear network coding. In this paper, we enhance the error correction capability of these two families of codes by using a novel two-tier decoding scheme. While the decoding of subspace and rank metric codes serves a second-tier decoding, we propose to perform a first-tier decoding on the packe…
Polar codes are able to achieve the capacity of memoryless channels under successive cancellation (SC) decoding. Soft Cancellation (SCAN) is a soft-output decoder based on the SC schedule, useful in iterative decoding and concatenation of polar codes. However, the sequential nature of this decoder leads to high decoding latency compared to state-of-the-art codes. To reduce the latency of SCAN, in this paper we identify special nodes in the decoding tree, corresponding to specific frozen-bit sequences, and propose dedicated low-latency decoding approaches for each of them. The resulting fast-SC…
This paper proposes a fountain coding system which has lower space decoding complexity and lower decoding erasure rate than the Raptor coding systems. The main idea of the proposed fountain code is employing shift and exclusive OR to generate the output packets. This technique is known as the zigzag decodable code, which is efficiently decoded by the zigzag decoder. In other words, we propose a fountain code based on the zigzag decodable code in this paper. Moreover, we analyze the overhead for the received packets, decoding erasure rate, decoding complexity, and asymptotic overhead of the pro…
Construction of PAC Codes with List-Search and Path-Splitting Critical Sets
针对高信噪比下 PAC 码偏离色散界及 SCL 解码排序开销大的问题，提出 LS 速率配置构造与 PSCS 方法，以改善 BLER 性能并减少排序操作。
PAC Learnability for Reliable Communication over Discrete Memoryless Channels
针对未知离散无记忆信道，提出基于PAC学习理论的数据驱动方法，通过虚拟样本算法选择解码度量和码率，实现接近信道互信息的可靠通信，证明DMCs的PAC可学习性。
RELDEC: Reinforcement Learning-Based Decoding of Moderate Length LDPC Codes
提出基于强化学习的RELDEC方法，通过MDP优化中等长度LDPC码的CN簇序列译码策略，并引入AM-RELDEC适应信道变化，性能显著优于传统方法。
In this paper we consider an interacting two-agent sequential decision-making problem consisting of a Markov source process, a causal encoder with feedback, and a causal decoder. Motivated by a desire to foster links between control and information theory, we augment the standard formulation by considering general alphabets and a cost function operating on current and previous symbols. Using dynamic programming, we provide a structural result whereby an optimal scheme exists that operates on appropriate sufficient statistics. We emphasize an example where the decoder alphabet lies in a space o…
A complexity-adaptive tree search algorithm is proposed for $\boldsymbol{G}_N$-coset codes that implements maximum-likelihood (ML) decoding by using a successive decoding schedule. The average complexity is close to that of the successive cancellation (SC) decoding for practical error rates when applied to polar codes and short Reed-Muller (RM) codes, e.g., block lengths up to $N=128$. By modifying the algorithm to limit the worst-case complexity, one obtains a near-ML decoder for longer RM codes and their subcodes. Unlike other bit-flip decoders, no outer code is needed to terminate decoding.…
Fast List Decoding of High-Rate Polar Codes
本文针对高速率极化码的SPC和SR1/SPC特殊节点提出两类快速列表译码算法，通过顺序译码和离线预确定候选路径降低延迟，仿真显示相比最先进快速SCL译码器延迟降低高达70.7%，且纠错性能不变。
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