Successive Cancellation List Decoding Research Articles

Neural network decoder of polar codes with tanh-based modified LLR over FSO turbulence channel.

The deep learning-based decoder of polar codes is investigated over free space optical (FSO) turbulence channel for the first time. The feedforward neural networks (NN) are adopted to establish the decoder and some custom layers are designed to train the NN decoder over the turbulence channel. The tanh-based modified log-likelihood ratio (LLR) is proposed as the input of NN decoder, which has faster convergence and better bit error rate (BER) performance compared with the standard LLR input. The simulation results show that the BER performance of NN decoder with tanh-based modified LLR is close to the conventional successive cancellation list (SCL) decoder over the turbulence channel, which verifies that the NN decoder with tanh-based modified LLR can learn the encoding rule of polar codes and the characteristics of turbulence channel. Furthermore, the turbulence-stability is investigated and the trained NN decoder in a fixed turbulence condition also has stable performance in other turbulence conditions.

An Efficient Software List Sphere Decoder for Polar Codes

Polar codes, first achieving the capacity of symmetric binary-input discrete memoryless channels (B-DMCs), have been standardized for eMBB control channels. Since 5G cellular requires flexible architecture which is realized by the software defined networking paradigm, efficient polar decoder is anticipated. Though successive cancellation list (SCL) decoder achieves satisfactory performance, it requires a large amount of memory. For short control channel codes, sphere decoder (SD) is an alternative, but costs unbearable time complexity at low signal-to-noise ratio. List sphere decoder (LSD) abandons the radius and keeps a list of best paths to gain a fixed complexity. However, LSD needs a large list size L for satisfactory performance. In this paper, an efficient software LSD with path pruning and efficient sorting is proposed. We recall the radius as the bound to delete the paths out of the sphere at very early levels. Since L is dynamic, efficient sorting is proposed to reduce the copy operations. Implemented with C++, the proposed decoder can reduce up to 65.3% latency compared with the original LSD, with the same performance and lower complexity.

Large-Small Sorting for Successive Cancellation List Decoding of Polar Codes

The successive cancellation list (SCL) decoding is used to achieve good error-correcting performance for practical finite-length polar codes. However, the metric sorting that is repeatedly performed in SCL decoding increases the overall decoding latency as the list size increases, making it difficult to apply the SCL decoding to the applications with limited processing time. To reduce the latency of the metric sorting, this paper proposes a new sorting method derived by analyzing path extension cases encountered in SCL decoding. The proposed method can avoid unnecessary sorting operations by adaptively determining the sorting size, and can be combined with other metric sorting methods. Simulation results show that the proposed method significantly reduces the decoding latency for various list sizes and code rates without degrading the error-correcting performance. The proposed method also becomes more effective as the code rate or the list size increases. Combined with the state-of-the-art sorting method, the proposed method reduces the average SCL decoding latency for a (1024, 512) polar code by 33.1% when the list size is 8.

Low-Complexity CRC Aided Joint Iterative Detection and SCL Decoding Receiver of Polar Coded SCMA System

As the fifth-generation (5G) wireless networks’ key technology, the joint design of SCMA and polar code is concerned by the future communication system. In this paper, a CRC aided joint iterative detection and successive cancellation list (SCL) decoding (CAJIDS) receiver is proposed for the polar coded SCMA (PC-SCMA) system. For the receiver, the SCL decoder’s extrinsic messages construction algorithm is designed by using Bayes rule and soft cancellation algorithm. Additionally, the distributed CRC aided polar (DCA-polar) code and variable list size are used to reduce the receiver’s decoding latency and complexity. Simulation results demonstrate that the CAJIDS receiver has better error rate performance than the joint iterative detection and decoding (JIDD) receiver. It also outperforms the LDPC coded SCMA (LDPC-SCMA) system. Specifically, when code length $N=256$ and code rate $R = 1/3$ , the CAJIDS outperforms the JIDD and LDPC-SCMA 1.2 dB and 1.9 dB over the Rayleigh fading channel, respectively. It also shows that, compared with the fixed-list-size receiver, with a similar error rate performance, the CAJIDS receiver has lower decoding latency and complexity.

An Area-Efficient Hybrid Polar Decoder With Pipelined Architecture

As the first kind of capacity-achieving forward error correction (FEC) codes, polar codes have attracted much research interest recently. Compared with traditional FEC codes, polar codes show better error correction performance when successive cancellation list (SCL) decoding with cyclic redundancy check is adopted. However, its serial decoding nature and high complexity of list management lead to its low throughput. Though the adaptive SCL decoding and hybrid decoding can improve the throughput, it comes at cost of implementation area. In this paper, we propose a pipelined hybrid decoding procedure and the corresponding hardware architecture to improve the area efficiency. In our design, the idle decoding cores are employed for successive cancellation (SC) decoding when SCL decoding is not working. The SCL decoding will be activated when the SC decoding fails. Different decoding cores work according to their own operation sequences and share one common processing array to improve the utilization ratio of processing elements. Constant receiving interval is supported with the design of input buffer to store all received codewords. A software platform is established to optimize the design parameters for each module of decoder. Moreover, the corresponding architecture is implemented using 65nm technology. Experimental results show that the proposed decoder can achieve a similar error correction performance with the SCL decoding with list size 16. Compared to the state-of-the-art available hybrid decoder, our proposed pipelined hybrid decoder is $3.07\times $ more area efficient.

Approaching the Normal Approximation of the Finite Blocklength Capacity within 0.025dB by Short Polar Codes

In this letter, we explore the performance limits of short polar codes and find that the maximum likelihood (ML) performance of a simple CRC-polar concatenated scheme can approach the normal approximation of the finite blocklength capacity. To explore the performance limits, CRC-polar concatenated codes are first optimized according to the minimum weight distribution. Then, CRC-aided hybrid decoding (CA-HD) algorithm with two steps is proposed to approach the ML performance. In the first step, the received sequence is decoded by the adaptive successive cancellation list (ADSCL) decoding. In the second step, the CRC bits of the survival paths in ADSCL are recalculated and CRC-aided sphere decoding with a reasonable initial radius calculated by the newly survival paths is used. The simulation results show that CRC-polar concatenated code with codeword length 128 and code rate 1/2 can achieve within about 0.025 dB of the normal approximation of the finite blocklength capacity at the block error rate 10−3.

Belief Propagation Decoder With Multiple Bit-Flipping Sets and Stopping Criteria for Polar Codes

Compared with successive cancellation list (SCL) decoders, belief propagation (BP)-based decoders suffer performance loss in middle-and high-signal-to-noise ratio (SNR) regions. By analyzing the behavior of the incorrect decoding results of the bit-flipping BP decoder with a critical set of order ω (BFBP-CS ω ), we found that undetected errors mainly contribute to the error floor. Based on this observation, we proposed a belief propagation decoder with multiple bit-flipping sets (BFSs) and stopping criteria (BP-MF-MC) in this work. We use multiple stopping criteria to identify undetected errors and a small BFS to find an additional estimated codeword given by the bit-flipping BP (BFBP) function. For uncorrected errors, we use multiple BFSs to find estimated codewords with the BFBP function. Furthermore, we propose a method to dynamically generate a BFS based on the submatrix check. This method can remove unnecessary bit-flipping positions and increase the order of the critical set. Then, the best codeword is selected from all estimated codewords according to the maximum likelihood principle. Numerical results show that BP-MF-MC performs similarly to the cyclic redundancy check-aided SCL (CA-SCL) decoder with list size 16 and is slightly worse than CA-SCL with list size 32.

An Adaptive Ordered Shifted-Pruning List Decoder for Polar Codes

The polar code has been selected as a channel coding solution for the control channel of 5th generation wireless systems (5G). Compared with the conventional cyclic-redundancy-check-aided successive cancellation list (CA-SCL) decoder for polar codes, the improved CA-SCL decoders not only can reduce the computational complexity but also keep similar block error rate (BLER). To further reduce the computational complexity and improve the BLER performance of the CA-SCL decoder, we propose an adaptive ordered shifted-pruning list (ADOSPL) decoder. The proposed method employs the ordered critical set (OCS) to improve the BLER performance and uses adaptive search width to reduce the computational complexity. Simulation results reveal that the ADOSPL decoder can significantly reduce 94.75% computational complexity compared with the CA-SCL decoder and improve the BLER performance gain up to 0.25dB for a polar code (512, 272) when the search width is 32. Moreover, lower computational complexity brings lower energy consumption and lower decoding delay in actual communication systems.

Practical Dynamic SC-Flip Polar Decoders: Algorithm and Implementation

SC-Flip (SCF) is a low-complexity polar code decoding algorithm with improved performance, and is an alternative to high-complexity (CRC)-aided SC-List (CA-SCL) decoding. However, the performance improvement of SCF is limited since it can correct up to only one channel error ($\omega=1$). Dynamic SCF (DSCF) algorithm tackles this problem by tackling multiple errors ($\omega \geq 1$), but it requires logarithmic and exponential computations, which make it infeasible for practical applications. In this work, we propose simplifications and approximations to make DSCF practically feasible. First, we reduce the transcendental computations of DSCF decoding to a constant approximation. Then, we show how to incorporate special node decoding techniques into DSCF algorithm, creating the Fast-DSCF decoding. Next, we reduce the search span within the special nodes to further reduce the computational complexity. Following, we describe a hardware architecture for the Fast-DSCF decoder, in which we introduce additional simplifications such as metric normalization and sorter length reduction. All the simplifications and approximations are shown to have minimal impact on the error-correction performance, and the reported Fast-DSCF decoder is the only SCF-based architecture that can correct multiple errors. The Fast-DSCF decoders synthesized using TSMC $65$nm CMOS technology can achieve a $1.25$, $1.06$ and $0.93$ Gbps throughput for $\omega \in \{1,2,3\}$, respectively. Compared to the state-of-the-art fast CA-SCL decoders with equivalent FER performance, the proposed decoders are up to $5.8\times$ more area-efficient. Finally, observations at energy dissipation indicate that the Fast-DSCF is more energy-efficient than its CA-SCL-based counterparts.

Enhanced Adaptive Successive-Cancellation List Decoder

Adaptive successive cancellation list decoder (ASCLD) with upsizing scheme as a well-known approach can significantly reduce the average list size of SCL algorithm for polar codes with cyclic redundancy check (CRC). Since it needs to re-decode the whole received bits if no survival path verifies CRC, it dramatically increases the decoding latency. In this paper, a segmented ASCLD is proposed to reduce the decoding delay of ASCLD, which deals with the multi-CRCs concatenated polar codes. Unlike the normal ASCLD, once all the survived paths of one decoding partition cannot pass its corresponding CRC, a new decoding attempt is immediately initiated with larger list size. Moreover, based on analyzing the performance of the proposed system, a series of optimization strategies are used to improve the performance. Simulation results show that the proposed scheme can greatly reduce the delay of upsizing ASCLD without performance loss.

An Adaptive Fusion Successive Cancellation List Decoder for Polar Codes with Cyclic Redundancy Check

An Adaptive Fusion Successive Cancellation List Decoder for Polar Codes with Cyclic Redundancy Check

Successive Cancellation List Decoding of Product Codes With Reed-Muller Component Codes

This letter proposes successive cancellation list (SCL) decoding of product codes with Reed–Muller (RM) component codes. SCL decoding relies on a product code description based on the $2\times 2$ Hadamard kernel, which enables interpreting the code as an RM subcode. The focus is on a class of product codes considered in wireless communication systems, based on single parity-check and extended Hamming component codes. For short product codes, it is shown that SCL decoding with a moderate list size performs as well as (and, sometimes, outperforms) belief propagation (BP) decoding. Furthermore, by concatenating a short product code with a high-rate outer code, SCL decoding outperforms BP decoding by up to 1.4 dB.

Belief Propagation Bit-Strengthening Decoder for Polar Codes

Although the performance of belief propagation (BP) decoder is slightly better than that of successive cancellation (SC) decoder, it has remarkable performance gap compared with improved SC decoders. In this letter, we propose BP bit-strengthening (BPBS) decoder and concept of relative code rate for polar codes. When conventional BP decoding fails, the stable information bits are found by submatrix check and relative code rate, then a priori message of stable information bits is strengthened by hard decision. The simulation results show that BPBS decoder can achieve significant performance gains compared with conventional BP decoder. Furthermore, as signal-to-noise ratio (SNR) increases, the performance of BPBS is close to that of successive cancellation list (SCL) decoder in the medium and high SNR regions.

Exploiting Error-Correction-CRC for Polar SCL Decoding: A Deep Learning-Based Approach

We investigate the cyclic redundancy check (CRC) codes aided successive cancellation list (SCL) decoding schemes to improve the performance of Polar codes. Distinguished from the existing literatures of Polar codes that only consider the error detection capability of CRC, we attempt to take advantage of the inherent error correction capability of CRC and we first devise a segmented CRC-error-correcting aided SCL decoding (SCC-SCL) framework. Based on this framework, an error-correcting table based SCC-SCL (ET-SCC-SCL) decoding scheme is proposed, by introducing the look-up-table based CRC error correction method into the segmented Polar SCL decoding process. Since the error correction capability is limited by the size of the look-up table, which in turn limits the performance gain, we further propose a deep learning based SCC-SCL (DL-SCC-SCL) decoding scheme. In this scheme, a long short-term memory (LSTM) network replaces the error correction table to perform error correction, combining the sequence of log likelihood ratios (LLRs) with the syndromes to determine the error patterns. Simulation results show that both of the proposed decoding schemes have significant performance gain over the classic CRC error-detection aided SCL decoding scheme. Especially for the DL-SCC-SCL, the performance gain at bit error rate of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-5</sup> is about 0.5 dB.

Hybrid Bucket Sorting Method for Successive Cancellation List Decoding of Polar Codes

This letter focuses on the path metric sorting of successive cancellation list (SCL) decoder for polar codes. The metric sorter is a major challenge of SCL decoder in terms of overall latency. We propose a hybrid bucket sorting method to select $L$ unordered paths with the smallest metrics out of the $2L$ paths. Each hybrid bucket sorter (HBS) consists of an optimized bucket sorter and a half cleaner. Evaluation results indicate that the HBS outperforms state-of-the-art metric sorters in terms of latency reduction.

Improved Adaptive Successive Cancellation List Decoding of Polar Codes

Although the adaptive successive cancellation list (AD-SCL) algorithm and the segmented-CRC adaptive successive cancellation list (SCAD-SCL) algorithm based on the cyclic redundancy check (CRC) can greatly reduce the computational complexity of the successive cancellation list (SCL) algorithm, these two algorithms discard the previous decoding result and re-decode by increasing L, where L is the size of list. When CRC fails, these two algorithms waste useful information from the previous decoding. In this paper, a simplified adaptive successive cancellation list (SAD-SCL) is proposed. Before the re-decoding of updating value L each time, SAD-SCL uses the existing log likelihood ratio (LLR) information to locate the range of burst error bits, and then re-decoding starts at the incorrect bit with the smallest index in this range. Moreover, when the segmented information sequence cannot get the correct result of decoding, the SAD-SCL algorithm uses SC decoding to complete the decoding of the subsequent segmentation information sequence. Furthermore, its decoding performance is almost the same as that of the subsequent segmentation information sequence using the AD-SCL algorithm. The simulation results show that the SAD-SCL algorithm has lower computational complexity than AD-SCL and SCAD-SCL with negligible loss of performance.

Analysis of the Block Error Probability of Concatenated Polar Code Ensembles

In this paper, we provide an analysis of the performance of concatenation of polar codes with outer cyclic redundancy check (CRC) codes, separated by an interleaver, in the short and moderate block length regimes. The analysis addresses maximum likelihood decoding as a proxy to the code performance under successive cancellation list decoding. The analysis is carried out by introducing the concatenated polar code (CPC) ensembles, whose distance properties can be analyzed (for sufficiently short block lengths) by means of the uniform interleaver approach. At moderate block lengths, we resort to the Monte Carlo simulations. Results show that if the inner polar code possesses a low minimum distance and the outer CRC code has a sufficiently large amount of redundancy, then the choice of the outer code generator polynomial and the interleaver may yield to a large variability in the performance of the resulting CPC.

Polar-Coded Physical Layer Network Coding Over Two-Way Relay Channels

Both polar codes and physical network coding (PNC) are promising techniques to achieve low-latency and high-rate transmission in 5G communications. In this letter, we investigate a joint polar-coded PNC system over a wireless two-way relay channel (TWRC). By exploiting the nature of PNC, we then propose two successive cancellation list (SCL) decoding schemes for the polar-coded PNC system, XOR-based conventional SCL (XOR-SCL) and joint decoding of PNC and SCL decoding (JPSCL), where message update rules are derived for odd and even channels. Simulation results demonstrate that JPSCL outperforms XOR-SCL over both AWGN and Rayleigh fading channels. In particular, JPSCL achieves a small bit error rate gap, i.e., within 1 dB at rate of 2/5, from SCL in the polar-coded point-to-point system at BER of $10^{-5}$ .

Improved Metric Sorting for Successive Cancellation List Decoding of Polar Codes

This letter focuses on the metric sorter, which is a module that affects the overall latency in the LLR-based successive cancellation list (SCL) decoder of polar codes. We propose two improved two-step sorters, which are achieved by revising the sorters used in the pruned bitonic extractor (PBE), and selecting one of two candidate architectures based on a threshold on $L$ . Moreover, we remove the first stage of the PBE in the second step sorting, which can reduce the latency in the second step and further increase the throughput of the sorting units. We first describe the architectures of the proposed sorters, and then derive the required number of the compare-and-swap units (CASUs) and the stages. The evaluation results indicate that the two proposed improved sorters outperform the state-of-the-art sorters in terms of the CASUs and the stages, which significantly lowers the overall latency and resource consumption. Compared with the PBE, the proposed sorters can reduce 57.8% of stages and 46.3% of CASUs for $L=128$ , and 33.3% of stages and 21.1% of CASUs for $L=8$ .

On Distance Properties of Convolutional Polar Codes

A lower bound on the minimum distance of convolutional polar codes is provided. The bound is obtained from the minimum weight of the generalized cosets of the codes generated by the bottom rows of the polarizing matrix. Moreover, a construction of convolutional polar subcodes is proposed, which provides improved performance under successive cancellation list decoding. For sufficiently large list size, the decoding complexity of convolutional polar subcodes appears to be lower compared with Arikan polar subcodes with the same performance. The error probability of successive cancellation list decoding of convolutional polar subcodes is lower than that of Arikan polar subcodes with the same list size.