Successive Cancellation List Research Articles

Neural Network Aided Path Splitting Strategy for Polar Successive Cancellation List Decoding

Combined with the path splitting selecting (PSS) strategy, the successive cancellation list (SCL) decoder achieves a lower decoding complexity without any performance loss. The main purpose of the PSS strategy is to select fewer non-frozen bits to perform the path splitting and pruning operations. To make the selection more efficient, a neural network aided path splitting selecting strategy is proposed in this paper. In the proposed method, the path splitting is seen as a classification problem. With the help of the neural network, the proposed method can precisely locate the non-frozen bits which are needed to be split. Simulation results verify that the proposed method can reduce the sorting complexity (which is equal to the number of path splitting) without any performance loss.

Low-Rate and Short-Block-Length Random Permutation-Coded Modulations Achieve Finite-Length Bounds

Ultra-reliable low-latency communication (URLLC) is an important feature brought by 5G New Radio (NR). By using a block code, the low-latency requirement in general requires a short block length in order to reduce the latency of transmitting the entire code block. The ultra-reliable communication requires a low-rate code to achieve high decoding reliability even under low signal-to-noise ratios (SNRs). This paper proposes a new class of coding and modulation schemes, termed permutation-coded modulations, for low-rate and short-block-length applications, such as URLLC. We show that permutation-coded modulations under maximum-likelihood (ML) decoding have remarkable performance levels that achieve the dispersion bounds with normal approximation (NA) for short block lengths. However, their encoding and ML-decoding complexities are prohibitive if the number of message bits transmitted per code block increases. To reduce the encoding and decoding complexities, we propose a trapezoidal permutation-coded modulation scheme which can be decoded efficiently by successive cancellation list (SCL) decoders. We also show that the trapezoidal permutation-coded modulations under SCL decoding can achieve the dispersion bounds with NA for short block lengths.

Judgement of error frames using frozen bits and its applications in decoding of polar codes

AbstractPolar codes have received widespread attention because of their excellent performance. However, although successive cancellation (SC)‐based decoding algorithms have achieved excellent error correction performance, they still encounter large time delays in decoding because a serial decoding structure is employed. At present, a cyclic redundancy check (CRC) is used to judge whether the decoding result of a frame is erroneous. To obtain such a result, the decoder has to wait until the end of decoding. In this work, a judgement strategy for error frames using frozen bits is proposed. With the proposed strategy, the recognition of erroneously decoded frames can be achieved based on the error state of the frozen bits in this frame. Accordingly, early termination of decoding can be realized. By using this mechanism, the strategy can also be applied in the SC List algorithm to prune the path. In addition, the SC Flip decoding algorithm can also use the proposed strategy to help find the first error message bit. The experimental results show that at least 85% of the frames decoded erroneously can be recognized by the proposed strategy so that decoding can be terminated in advance and computational resources can be saved.

Decoding of the extended Golay code by the simplified successive-cancellation list decoder adapted to multi-kernel polar codes

This paper describes an adaptation of a polar code decoding technique in favor of the extended Golay code. Based on the bridge provided by a permutation matrix between the code words of these two classes of codes, the Golay code can be decoded by any polar code technique. Contrary to the successive-cancellation list technique which is characterized by a serial estimation of the bits, we propose in this work an adaptation of the simplified successive-cancellation list technique to polar codes equivalent to the Golay code. The simulations have achieved the performance of a maximum likelihood decoding, with the low decoding complexity of polar codes, compared to one of the universal decoders of linear codes most known in the literature.

The implementation of Shannon-limited polar codes-based information reconciliation for quantum key distribution

Quantum key distribution (QKD) gives a way to generate unconditionally secure keys for two remote users, Alice and Bob. Information reconciliation (IR), which can correct the errors caused by the imperfections of the QKD systems, is a critical component in QKD. Due to the high-security requirements and large volumes of data processing, robustness and efficiency are two main factors that must be considered for the implementation of IR. The polar codes-based IR has several potential advantages, such as capacity to reach Shannon-limit, high IR efficiency, and low computational complexity. Although CPU-based IR is always implemented in most of the previous works, it is not the optimal implementation in terms of performance and power dissipation. To the best of our knowledge, there is still no work to build a special-purpose hardware module for polar codes-based IR. In this paper, a dedicated design of hardware accelerator is first proposed for polar codes-based IR, in which the block-checked successive cancellation list (SCL) algorithm is used to verify the consistency of the sifted keys, and the overall failure probability of IR is significantly reduced. The proposed design is constructed into a partially-unrolled parallel architecture to accelerate the core decoder as well as balance the resource utilization. Furthermore, the hardware implementation is completed based on Xilinx Zynq UltraScale+ XCZU5EV MPSoC platform and achieves an IR throughput of 15 Mbps with a block length of 212, while less than of the amount of on-chip resources are used in other previous designs of SCL decoder. The proposed design can provide a real-time, low-cost solution for IR in QKD systems, and enhance the performance of QKD.

A Tree Pruning Technique for Decoding Complexity Reduction of Polar Codes and PAC Codes

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 visible degradation in 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 metric value deviates from the bit-channel capacity’s partial summation. For relatively reliable bit-channels, the bit metric for a wrong branch becomes very large negative number, which enables us to detect and prune such paths. We prove that, for a threshold lower than the bit-channel cutoff rate, the probability of pruning the correct path decreases exponentially by the given threshold. Based on these findings, we presented a pruning technique, and the experimental results demonstrate a substantial decrease in the amount of sorting procedures required for SCL decoding. In the stack algorithm, a similar technique is used to significantly reduce the average number of paths in the stack.

On the improvements of successive cancellation Creeper decoding for polar codes

In this paper, an improvement for successive cancellation (SC) decoding is presented for polar codes. This improvement is based on applying Creeper approach that was used to decode convolutional codes. The proposed decoder combines a stack and a Fano approaches, allowing to significantly decrease complexity of Successive Cancellation Fano decoder at the cost of a slight increase in space complexity. We also show how to combine the SC-Creeper decoding with a bit-flipping approach to further improve the performance of SC-Creeper due to introducing additional decoding attempts.Simulation results show that the proposed SC-Creeper and SC-Creeper Flip decoders with carefully selected parameters have the same performance as both CRC-aided Successive Cancellation List decoder with list size 8 and Fano decoder, while having much smaller complexity.

Application research of polar coded OFDM underwater acoustic communications

To further accelerate the practical process of polar codes in underwater acoustic communication (UWC) and improve the reliability and efficiency of underwater data transmission, we first propose a polar code construction method suitable for the underwater acoustic channel, which drastically reduces the complexity and meets the need for practical UWC. Then, we establish a practical polar coded UWC scheme by devising a two-step procedure based on the Orthogonal Frequency Division Multiplexing technique. Moreover, through simulation, the scheme's performances with the unimproved and the proposed polar code construction method are analyzed. Moreover, its performances with four different polar decoders are also given. The performance comparison with the LDPC coded UWC system is studied as well. Furthermore, lake-trial results under two different channel conditions demonstrate that the proposed construction method and the scheme effectively guarantee the reliability of data transmission, achieving error-free transmission at a transmission distance of 1718 m with the signal-to-noise-ratio (SNR) of 21 dB, and a transmission distance of 749 m with the SNR of 16.5 dB, better than the LDPC coded system with same conditions. The semi-experimental results show that the proposed polar coded system under the Cyclic Redundancy Check-Aided Successive Cancellation List decoder outperforms the LDPC coded system by approximately 2.1–3.7 dB.

Wireless Covert Communication with Polarization Dirty Constellation

Wireless covert communication is an emerging communication technique that prevents eavesdroppers. This paper presents a novel approach to wireless covert communication based on polar codes with dirty constellation polarization. The feasibility of dirty constellation polarization is analyzed, and the impact of covert transmission power on undetectability is investigated. We polarize the covert channel and decode the covert information using a successive-cancellation decoder. Simulation results demonstrate that polar codes can significantly improve the undetectability and reliability of covert communication when compared to BCH codes. Additionally, we find that polarized covert channels achieve better undetectability and reliability when using a successive-cancellation list decoder. The proposed algorithm shows significant improvement in the undetectability and reliability of wireless covert communication with dirty constellations. This research provides a promising direction for future development in the field of covert communication, with potential applications in secure communication systems.

List-serial pipelined hardware architecture for SCL decoding of polar codes

For polar codes, the performance of successive cancellation list (SCL) decoding is capable of approaching that of maximum likelihood decoding. However, the existing hardware architectures for the SCL decoding suffer from high hardware complexity due to calculating L decoding paths simultaneously, which are unfriendly to the devices with limited logical resources, such as field programmable gate arrays (FPGAs). In this paper, we propose a list-serial pipelined hardware architecture with low complexity for the SCL decoding, where the serial calculation and the pipelined operation are elegantly combined to strike a balance between the complexity and the latency. Moreover, we employ only one successive cancellation (SC) decoder core without L×L crossbars, and reduce the number of inputs of the metric sorter from 2L to L + 2. Finally, the FPGA implementations show that the hardware resource consumption is significantly reduced with negligible decoding performance loss.

Low-Latency SCL Polar Decoder Architecture Using Overlapped Pruning Operations

Allowing the superior error-correction performance even for short-length codewords, the successive-cancellation list (SCL) decoding algorithm has allowed the polar code to be adopted in 5G New Radio standard for control channel. However, existing SCL polar decoders still suffer from long processing latency caused by a number of serialized internal operations. In this work, to solve the latency problem, we present several parallel computing solutions for the serialized operations, i.e., simplified data dependencies and two overlapped pruning operations. To realize the proposed parallel computing, we also introduce internal circuit blocks including dual read-port buffers, an on-the-fly parity checker, and overlapped processing units. The proposed SCL polar decoders are precisely designed with optimal design parameters by analyzing trade-offs between the latency reduction and area overheads. Implemented in a 65-nm CMOS technology, the proposed list-8 SCL polar decoder requires only 374 ns to handle a (1024, 512) 5G codeword, improving the decoding efficiency by 34.7% compared to the previous designs.

Optimized Higher-Order Polarization Weight Incremental Selective Decoding and Forwarding in Cooperative Satellite Sensor Networks

An optimized higher-order polarization weight (HPW) incremental selective decoding and forwarding (ISDF) scheme is proposed for cooperative transmissions in satellite sensor networks (SSNs). First, an HPW method is designed with subchannel power allocation in encoding to enhance the antiinterference. Second, a cooperative decision in the destination is used to avoid error propagation for better outage probability (OP) and bit error rate (BER) performance. It switches between direct and cooperative transmissions alternatively by channel state information (CSI). Finally, the HPW coding information, cyclic redundancy codes (CRCs), and four special frame structures of polar subcodes are jointly fast successive cancellation list (FSCL) decoded to improve efficiency and BER performance. Finally, we apply the optimized polarization encoding and decoding to the relay system to form a new ISDF relay system for better coding efficiency, BER, and OP performance. Simulation results show that the proposed scheme obtains approximately 0.8 and 1 dB performance gains at OP of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${{10}^{-{2}}}$ </tex-math></inline-formula> and BER of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${{10}^{-{6}}}$ </tex-math></inline-formula> , respectively, when compared with those of the current ISDF scheme. Therefore, the cooperative scheme possesses efficient decoding and cooperation, which endows it a promising candidate relay scheme in SSNs.

A golden decade of polar codes: From basic principle to 5G applications

After the pursuit of seventy years, the invention of polar codes indicates that we have found the first capacity-achieving coding with low complexity construction and decoding, which is the great breakthrough of the coding theory in the past two decades. In this survey, we retrospect the history of polar codes and summarize the advancement in the past ten years. First, the primary principle of channel polarization is investigated such that the basic construction, coding method and the classic successive cancellation (SC) decoding are reviewed. Second, in order to improve the performance of the finite code length, we introduce the guiding principle and conclude five design criteria for the construction, design and implementation of the polar code in the practical communication system based on the exemplar schemes in the literature. Especially, we explain the design principle behind the concatenated coding and rate matching of polar codes in 5G wireless system. Furthermore, the improved SC decoding algorithms, such as SC list (SCL) decoding and SC stack (SCS) decoding etc., are investigated and compared. Finally, the research prospects of polar codes for the future 6G communication system are explored, including the optimization of short polar codes, coding construction in fading channels, polar coded modulation and HARQ, and the polar coded transmission, namely polar processing. Predictably, as a new coding methodology, polar codes will shine a light on communication theory and unveil a revolution in transmission technology.

Simplified Metric Sorting for Multi-Bit Successive Cancellation List Decoding of Polar Codes

This letter focuses on the metric sorting step for multi-bit successive cancellation list (SCL) decoders of polar codes, which is an important part of the whole decoder. We propose a primary-secondary exchange sorting (PSES) method. In the primary sorting, most of the survivals, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${L}$ </tex-math></inline-formula> paths are selected from the first minor <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2L</i> candidates. In the secondary sorting, a small number of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${N}$ </tex-math></inline-formula> paths are extracted from the other huge number of candidates to remedy the missing survivals. The <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${N}$ </tex-math></inline-formula> paths in the secondary stage are exchanged with the largest <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${N}$ </tex-math></inline-formula> elements in the first <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${L}$ </tex-math></inline-formula> paths. Due to the minor candidates in the primary stage and the few survivals in the secondary stage, this method reduces the complexity with negligible performance loss. Additionally, a sorting network based on compare-and-swap unit (CASU) PSES (CPSES) and a sorting network based on Radix-2L PSES (RPSES) are proposed to improve secondary sorting. The evaluation results show that compared with state-of-the-art metric sorting methods, the proposed CPSES architecture significantly reduces hardware resources, and the proposed RPSES architecture significantly reduces the latency.

Successive Cancellation Decoding of Single Parity-Check Product Codes: Analysis and Improved Decoding

A product code with single parity-check component codes can be described via the tools of a multi-kernel polar code, where the rows of the generator matrix are chosen according to the constraints imposed by the product code construction. Following this observation, successive cancellation decoding of such codes is introduced. In particular, the error probability of single parity-check product codes over binary memoryless symmetric channels under successive cancellation decoding is characterized. A bridge with the analysis of product codes introduced by Elias is also established for the binary erasure channel. Successive cancellation list decoding of single parity-check product codes is then described. For the provided example, simulations over the binary input additive white Gaussian channel show that successive cancellation list decoding outperforms belief propagation decoding applied to the code graph. Finally, the performance of the concatenation of a product code with a high-rate outer code is investigated via distance spectrum analysis. Examples of concatenations performing within 0.7 dB from the random coding union bound are provided.

Single parity check node adapted to polar codes with dynamic frozen bit equivalent to binary linear block codes

&lt;span lang="EN-US"&gt;In the context of decoding binary linear block codes by polar code decoding techniques, we propose in this paper a new optimization of the serial nature of decoding the polar codes equivalent to binary linear block codes. In addition to the special nodes proposed by the simplified successive-cancellation list technique, we propose a new special node allowing to estimate in parallel the bits of its sub-code. The simulation is done in an additive white gaussian noise channel (AWGN) channel for several linear block codes, namely bose–chaudhuri–hocquenghem codes (BCH) codes, quadratic-residue (QR) codes, and linear block codes recently designed in the literature. The performance of the proposed technique offers the same performance in terms of frame error rate (FER) as the ordered statistics decoding (OSD) algorithm, which achieves that of maximum likelihood decoder (MLD), but with high memory requirements and computational complexity.&lt;/span&gt;

Intelligent Path-Selection-Aided Decoding of Polar Codes

CRC-aided successive cancellation list (CA-SCL) decoding is a powerful algorithm that dramatically improves the error performance of polar codes. Path selection is a major issue that affects the decoding latency of SCL decoders. Generally, path selection is implemented using a metric sorter, which causes its latency to increase as the list grows. In this paper, intelligent path selection (IPS) is proposed as an alternative to the traditional metric sorter. First, we found that in the path selection, only the most reliable paths need to be selected, and it is not necessary to completely sort all paths. Second, based on a neural network model, an intelligent path selection scheme is proposed, including a fully connected network construction, a threshold and a post-processing unit. Simulation results show that the proposed path-selection method can achieve comparable performance gain to the existing methods under SCL/CA-SCL decoding. Compared with the conventional methods, IPS has lower latency for medium and large list sizes. For the proposed hardware structure, IPS's time complexity is O(klog2(L)) where k is the number of hidden layers of the network and L is the list size.

Polar-code-based information reconciliation scheme with the frozen-bit erasure strategy for quantum key distribution

Information reconciliation (IR) ensures the correctness of quantum key distribution systems, by correcting the error bits existed in the sifted keys. In this article, we propose a polar codes-based IR scheme with the frozen bits erasure strategy, where an equivalent transmission of sifted keys is conducted, so that the frozen bits in the decoding procedure is erased to 0. Thus, our IR scheme can be implemented efficiently without the assumption of true random numbers. Furthermore, we implement the proposed IR scheme with the fast simplified successive cancellation list decoder and its throughput reaches to 0.88Mbps with the yield of 0.8333, where the decoder list size is 16, the block size is 1Mb and the quantum bit error rate is 0.02.

Deep learning aided SCL decoding of polar codes with shifted-pruning

Recently, a generalized successive cancellation list (SCL) decoder implemented with shifted-pruning (SP) scheme, namely the SCL-SP-ω decoder, is presented for polar codes, which is able to shift the pruning window at most ω times during each SCL re-decoding attempt to prevent the correct path from being eliminated. The candidate positions for applying the SP scheme are selected by a shifting metric based on the probability that the elimination occurs. However, the number of exponential/logarithm operations involved in the SCL-SP-ω decoder grows linearly with the number of information bits and list size, which leads to high computational complexity. In this paper, we present a detailed analysis of the SCL-SP-ω decoder in terms of the decoding performance and complexity, which unveils that the choice of the shifting metric is essential for improving the decoding performance and reducing the re-decoding attempts simultaneously. Then, we introduce a simplified metric derived from the path metric (PM) domain, and a custom-tailored deep learning (DL) network is further designed to enhance the efficiency of the proposed simplified metric. The proposed metrics are both free of transcendental functions and hence, are more hardware-friendly than the existing metrics. Simulation results show that the proposed DL-aided metric provides the best error correction performance as comparison with the state of the art.

Joint Design of Permutations and Polar Code Under BPL Decoding

Recently, belief propagation list (BPL) decoding of polar codes has been proposed for achieving state-of-the-art performance. By representing a polar code factor graph permutation with an equivalent bit index permutation, a single decoder can be applied on different permutated factor graphs, which is hardware-friendly. Nevertheless, the design of permutations under BPL decoding turns out to be cumbersome and highly relevant with the code structure. In this work, we propose a two-step approach for the joint design of permutations and polar code under BPL decoding. Firstly, an empirical approach is proposed for selection of good initial permutations under BPL decoding. Then, the joint optimization of permutations and polar code is implemented based on genetic algorithm. With the aid of an outer cyclic redundancy check (CRC) code, the proposed BPL scheme is shown to perform very close to CRC aided successive cancellation list (CA-SCL) decoding at practical SNRs.