CRC-aided Successive Cancellation List Research Articles

Chaotic Phase Modulation Direct-Sequence Spread Spectrum-Assisted Adaptive Serial Cancellation List Decoding Method for Underwater Acoustic Communication

Addressing the challenges of high decoding latency, reduced spectral efficiency, and substantial storage requirements in a Cyclic Redundancy Check (CRC)Aided Successive Cancellation List (CA-SCL) polar decoder, this paper proposes a chaotic phase modulation direct-sequence spread spectrum (CPMDSSS)-assisted adaptive serial cancellation list decoding method for underwater acoustic communication. The method involves segmenting the information bits to be transmitted, computing CRC for each segment, and mapping all CRCs to a CPMDSSS, which is then modulated onto the pilot subcarriers of underwater acoustic orthogonal frequency-division multiplexing (OFDM) to increase spectral efficiency. At the receiving end, CRCs are obtained by demodulating the CPMDSSS to verify the segmented information and adaptively select the number of decoding paths. The theoretical analysis and simulation results demonstrate that compared to CA-SCL, the proposed method effectively reduces the required storage units and improves spectral efficiency, with an average reduction of approximately 80% in the decoding paths. Sea trials further indicate that the proposed method reduces the average decoding paths by approximately 71% and decreases the average decoding delay by approximately 64% compared to CA-SCL.

Efficient protection of polar decoders against Single Event Upsets (SEUs) on user memories

Polar codes are used in 5G systems for the transmission of control channels due to their excellent error correction capability for short sequences. CRC aided successive cancellation list (CA-SCL) decoders are commonly used in practical systems. When used in critical environments, e.g. space platforms, memories used in the polar decoder will suffer single-event upsets (SEUs) that can cause failures and disrupt communications. Therefore, the protection of polar decoders against SEUs on memories with low overhead is an important problem, especially for resource-limited on-board space systems. In this paper, efficient protection schemes are first proposed for each of the main memory components of CA-SAL decoders based on their algorithmic properties. Then FPGA implementation and hardware-based fault injection experiments are performed to evaluate the protection overhead and reliability of the proposed schemes. Experimental results show that, relative to the unprotected decoder, the reliability of the decoder is improved by more than two orders of magnitude, and the overhead of memory and computation are only about 1.26 and 1.23 times of those of unprotected decoder.

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.

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.

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.

Adaptive Belief Propagation Decoding of CRC Concatenated NR LDPC and Polar Codes

In this paper, we propose a modified adaptive belief propagation (ABP) algorithm, which is referred to as the random ABP (R-ABP) algorithm, for decoding short channel codes adopted in the fifth generation (5G) New Radio (NR) wireless systems. Based on the cyclic redundancy check (CRC) concatenation structure of 5G channel codes, we can take partial or even all CRC bits into iterative R-ABP decoding to improve the error correction performance, while the error detection capability can still be guaranteed by the remaining CRC bits and a proposed threshold-based acceptance criterion. We call this improved R-ABP decoding the threshold-and-CRC-aided R-ABP (TCA-R-ABP) decoding. The simulation results show that our proposed TCA-R-ABP algorithm outperforms the state-of-the-art decoding algorithms for many 5G low-density parity-check (LDPC) and polar codes. Moreover, the proposed TCA-R-ABP algorithm leads to a unified decoder for LDPC and polar codes, which has the potential to be a lower-complexity decoder over the combination of the belief propagation (BP) and CRC-aided successive cancellation list (CA-SCL) decoders.

Performance of Polar Codes over Generalized Correlated Fading Channels

Polar coding, as introduced by Arikan in 2009, is an error correcting code scheme that uses the polarization technique to obtain almost noise-free channels that are suitable for transmission. Polar codes are being widely studied due to their applications on the 5th Generation (5G) of mobile communication systems. In 5G communications, the wireless channel is subject to fading that can significantly degrade the quality of the transmitted signals. In this context, the generalized fading models are useful due to their capacity to model a variety of channel conditions ranging from moderate to severe fading by adjusting some parameters. In this work, the authors investigate the performance of polar codes considering the transmission over generalized fading channels. Additionally, the effect of channel correlation is analyzed and a random interleaver is proposed to mitigate the negative impact of the correlation on the system performance. For the system with generalized fading modeled by eta-mu distribution and Cyclic Redundancy Check (CRC) Aided Successive Cancellation List (CA-SCL) decoding, the scenario with correlated fading channel and interleaving shows a performance improvement compared to the system without interleaving. In this case, a gain of approximately 3.5 dB is obtained considering a Bit Error Rate (BER) of 10^{-2}. Considering the alpha-mu (alpha=2 and mu=1) and the kappa-mu  (kappa=mu=2) fading channels,  the uncorrelated systems outperform the correlated ones. Among them, the interleaver improves the performance of the systems. For a BER of 10^{-2}, as an example, the use of fading correlated channels and interleaver shows a performance improvement of 12 dB and 0.5 dB for the alpha-mu and kappa-mu fading channels, respectively.

An Enhanced Belief Propagation Decoder for Polar Codes

Belief propagation (BP) has attracted increasing attention due to its high parallelism. In this letter, a correction method is proposed to improve the block error rate (BLER) performance of the BP decoder for polar codes so that it is comparable to that of the successive cancellation list (SCL) decoder. Compared with BP flipping (BPF) decoding, the proposed decoder focuses on prior knowledge of the code bits in the factor graph. Two strategies have been proposed to construct the correction set. The simulation results show that the BPC decoder exhibits a BLER similar to that of BPF decoder while reducing the average latency and complexity by half. In addition, the BLER of the BPC decoder can approach that of the cyclic redundancy check (CRC)-aided SCL (CA-SCL) decoder in regions with low and medium signal-to-noise ratios (SNRs).

IM/DD probabilistically shaped 64-QAM OFDM-PON transmission assisted by selective-mapping and flexible systematic polar code

Intensity-modulation direct-detection (IM/DD) orthogonal frequency division multiplexing (OFDM) has advantages of low cost and simple structure, which is a promising candidate for next-generation high-speed passive optical networks. In this work, we propose and experimentally demonstrate a look-up-table-based probabilistic shaping scheme assisted by selective-mapping (SLM) and flexible systematic polar code (SPC) to improve the performance of IM/DD 64-QAM OFDM transmission system. Meanwhile, we experimentally compare the performance between the proposed scheme and the joint constant composition distribution matching (CCDM) and low-density parity-check (LDPC). Here, abundant phase sequences in the SLM are generated by chaotic sequence to suppress peak-to-average power ratio (PAPR). The pre-set bits of SPC are utilized for the additional information of probabilistic shaping to reduce the redundancy and the cyclic redundancy check (CRC) aided successive cancellation list (CA-SCL) decoder is used for decoding. Joint probabilistic shaping and PAPR suppression have contributed 4.6-dB and 3-dB bit error ratio performance improvements for the IM/DD 64-QAM OFDM transmission system in the pre and post forward error correction (FEC), respectively.

Fast Frequency Sweep Analysis of Passive Miniature RF Circuits Based on Analytic Extension of Eigenvalues

The construction of polar codes with code length $n=2^m$ involves $m$ layers of polar transforms. In this paper, we observe that after each layer of polar transforms, one can swap certain pairs of adjacent bits to accelerate the polarization process. More precisely, if the previous bit is more reliable than its next bit under the successive decoder, then switching the decoding order of these two adjacent bits will make the reliable bit even more reliable and the noisy bit even noisier. Based on this observation, we propose a new family of codes called the Adjacent-Bits-Swapped (ABS) polar codes. We add a permutation layer after each polar transform layer in the construction of the ABS polar codes. In order to choose which pairs of adjacent bits to swap in the permutation layers, we rely on a new polar transform that combines two independent channels with $4$-ary inputs. This new polar transform allows us to track the evolution of every pair of adjacent bits through different layers of polar transforms, and it also plays an essential role in the Successive Cancellation List (SCL) decoder for the ABS polar codes. Extensive simulation results show that ABS polar codes consistently outperform standard polar codes by 0.15dB--0.3dB when we use CRC-aided SCL decoder with list size $32$ for both codes. The implementations of all the algorithms in this paper are available at https://github.com/PlumJelly/ABS-Polar

Convolutional Neural Network-Aided Tree-Based Bit-Flipping Framework for Polar Decoder Using Imitation Learning

Known for their capacity-achieving abilities and low complexity for both encoding and decoding, polar codes have been selected as the control channel coding scheme for 5G communications. To satisfy the needs of high throughput and low latency, belief propagation (BP) is chosen as the decoding algorithm. However, it suffers from worse error performance than that of cyclic redundancy check (CRC)-aided successive cancellation list (CA-SCL). Recently, convolutional neural network-aided bit-flipping (CNN-BF) is applied to BP decoding, which can accurately identify the erroneous bits to achieve a better error rate and lower decoding latency than prior critical-set bit-flipping (CS-BF) mechanism. However, successive BF, having better error correction capability, has not been explored in CNN-BF since the more complicated flipping strategy is out of the scope of supervised learning. In this work, by using imitation learning, a convolutional neural network-aided tree-based multiple-bits BF (CNN-Tree-MBF) mechanism is proposed to explore the benefits of multiple-bits BF. With the CRC information as additional input data, the proposed CNN-BF model can further reduce 5 flipping attempts. Besides, a tree-based flipping strategy is proposed to avoid useless flipping attempts caused by wrongly flipped bits. From the simulation results, our approach can outperform CS-BF and reduce flipping attempts by 89% when code length is 64, code rate is 0.5 and SNR is 1 dB. It also achieves a comparable block error rate (BLER) as CA-SCL.

Post-Processing for CRC-Aided Successive Cancellation List Decoding of Polar Codes

In this letter, three post-processing schemes for cyclic redundancy check (CRC-)aided successive cancellation list (CA-SCL) decoding are presented for polar codes. The re-decoding based on our post-processing schemes can effectively alleviate both the channel and decision errors encountered via CA-SCL decoding. Simulation results show that our schemes perform better than conventional re-decoding schemes and can be incorporated into a polar decoder that has a small list size to achieve a performance that is expected for a large list size.

Deep Learning-Aided Belief Propagation Decoder for Polar Codes

This paper presents deep learning (DL) methods to optimize polar belief propagation (BP) decoding and concatenated LDPC-polar codes. First, two-dimensional offset Min-Sum (2-D OMS) decoding is proposed to improve the error-correction performance of existing normalized Min-Sum (NMS) decoding. Two optimization methods used in DL, namely back-propagation and stochastic gradient descent, are exploited to derive the parameters of proposed algorithms. Numerical results demonstrate that there is no performance gap between 2-D OMS and exact BP on various code lengths. Then the concatenated OMS algorithms with low complexity are presented for concatenated LDPC-polar codes. As a result, the optimized concatenated OMS decoding yields error-correction performance with CRC-aided successive cancellation list (CA-SCL) decoder of list size 2 on length-1024 polar codes. In addition, the efficient hardware architectures of scalable polar OMS decoder are described and the proposed decoder is reconfigurable to support three code lengths ( $N= 256, 512, 1024$ ) and two decoding algorithms (2-D OMS and concatenated OMS). The polar OMS decoder implemented on 65 nm CMOS technology achieves a maximum coded throughput of 5.4 Gb/s at $E_{b}/N_{0} = 4$ dB for code length 1024 and 7.5 Gb/s at $E_{b}/N_{0} = 3.5$ dB for code length 256, which are comparable to the state-of-the-art polar BP decoders. Moreover, a 5.1 Gb/s throughput at $E_{b}/N_{0} = 4$ dB is achieved under concatenated OMS decoding mode for code length 1024 with a latency of 200 ns, which is superior to existing CA-SCL decoders that have similar error-correction performance.

A Node-Reliability Based CRC-Aided Successive Cancellation List Polar Decoder Architecture Combined With Post-Processing

This paper presents a node-reliability based cyclic redundancy check (CRC)-aided successive cancellation list (NRB CA-SCL) polar decoder architecture, where node reliability is used to determine the number of expanding paths, and a priority-based sorter is devised so as to reduce the number of sorting cycles. In order to reduce the effort required for sorting by the node processor, the expanding segments are generated using off-line flipping patterns. We also propose a post-processing scheme that improves the error-rate performance for the NRB CA-SCL decoders. The proposed techniques are demonstrated by implementing (1024, 512) polar decoders. Compared to other state-of-the-art CA-SCL decoders, the proposed decoder, when the post-processing technique is not included, achieves a better area efficiency. When the post-processing technique is included, the proposed list-16 decoder achieves an area efficiency that is almost 2.5 times greater compared to the latest list-32 CA-SCL decoder, as well as a similar error-rate performance.

Decoder-Tailored Polar Code Design Using the Genetic Algorithm

We present a new framework for constructing polar codes (i.e., selecting the frozen bit positions) for arbitrary channels, tailored to a given decoding algorithm rather than assuming the (not necessarily optimal) successive cancellation (SC) decoding. The proposed framework is based on the genetic algorithm (GenAlg), where populations (i.e., collections) of information sets evolve via evolutionary transformations based on their individual error-rate performance. These populations converge toward an information set that fits both the decoding behavior and the defined channel. We construct polar codes, without the CRC-aid, tailored to plain successive cancellation list (SCL) decoding, achieving the same error-rate performance as the CRC-aided SCL decoding over both the AWGN channel and the Rayleigh channel, respectively. Furthermore, a proposed belief propagation (BP)-tailored construction approaches the SCL error-rate performance without any modifications in the decoding algorithm itself. The performance gains can be attributed to the significant reduction in the number of low-weight codewords. We show that, when required, the GenAlg can also be set up to find codes that reduce the decoding complexity. This way, the SCL list size or the number of BP iterations can be reduced while maintaining the same error-rate performance.

Sphere decoder for polar codes concatenated with cyclic redundancy check

The existing cyclic redundancy check (CRC)-aided successive cancellation list (CA-SCL) decoder partitions the decoding process into two steps,where an SCL is followed by a CRC check. An SCL decoder can approach the maximum-likelihood (ML) decoding performance of the inner polar codes using a sufficiently large list; however, in this case, CRC is only used for performing error detection. Therefore, the decoding performance of the outer CRC is different from that of ML because the errors are not rectified, which degrades the performance of the entire concatenated codes. In this study, we propose a sphere decoder (SD)that can achieve the ML performance of polar codes concatenated with CRC to address the suboptimality of CA-SCL decoding. The proposed SD performs joint decoding of the CRC-polar codes in a single step, thereby avoiding the polar decoding and CRC detection decoding scheme. Because the proposed SD guarantees the ML decoding performance of the CRC-polar concatenated codes, the simulation results demonstrate that the block error rate (BLER) of the proposed SD acts as the lower bound of the CA-SCL decoding performance. Further, a new initial radius selection method is proposed for the SD to reduce the average decoding complexity. The simulations demonstrate that the proposed initial radius selection method reduces more amount of decoding complexity when compared with that reduced using sequential step size methods.

CRC-Aided Sphere Decoding for Short Polar Codes

Polar codes have recently been adopted as a coding scheme for fifth-generation wireless systems, owing to their excellent performance at short code lengths. In this letter, we propose a cyclic redundancy check (CRC)-aided sphere decoding (CA-SD) algorithm to further improve the performance of short polar codes. It provides better performance than the widely used CRC-aided successive cancellation list (CA-SCL) decoding, and it is robust to the CRC length that leads to the stable performance under a wide range of code rates. By employing the Gaussian elimination method, we transform the underlying parity check relationships in CRC as the equivalent new forms. This ensures the parity check bits can be uniquely judged by previous decoded bits rather than the redundant search during the CA-SD. By this means, the complexity of CA-SD is reduced and comparable with the CA-SCL decoding under short code lengths.

Segmented Successive Cancellation List Polar Decoding with Tailored CRC

As the first error correction codes provably achieving the symmetric capacity of binary-input discrete memory-less channels (B-DMCs), polar codes have been recently chosen by 3GPP for eMBB control channel. Among existing algorithms, CRC-aided successive cancellation list (CA-SCL) decoding is favorable due to its good performance, where CRC is placed at the end of the decoding and helps to eliminate the invalid candidates before final selection. However, the good performance is obtained with a complexity increase that is linear in list size $L$. In this paper, the tailored CRC-aided SCL (TCA-SCL) decoding is proposed to balance performance and complexity. Analysis on how to choose the proper CRC for a given segment is proposed with the help of \emph{virtual transform} and \emph{virtual length}. For further performance improvement, hybrid automatic repeat request (HARQ) scheme is incorporated. Numerical results have shown that, with the similar complexity as the state-of-the-art, the proposed TCA-SCL and HARQ-TCA-SCL schemes achieve $0.1$ dB and $0.25$ dB performance gain at frame error rate $\textrm{FER}=10^{-2}$, respectively. Finally, an efficient TCA-SCL decoder is implemented with FPGA demonstrating its advantages over CA-SCL decoder.

Belief Propagation Bit-Flip Decoder for Polar Codes

The bit-flip method has been successfully applied to the successive cancellation (SC) decoder to improve the block error rate (BLER) performance for polar codes in the finite code length region. However, due to the sequential decoding, the SC decoder inherently suffers from longer decoding latency than that of the belief propagation (BP) decoder with efficient early stopping criterion. It is natural to ask how to perform bit-flip in a polar BP decoder. In this paper, bit-flip is introduced into the BP decoder for polar codes. The idea of critical set (CS), that is, originally proposed by Zhang et al. for identifying unreliable bits in a SC bit-flip decoder, is extended to the BP decoder here. After revealing the relationship between CS and the incorrect BP decoding results, critical set with order ω (CS-ω) is constructed to identify unreliable bit decisions in polar BP decoding. The simulation results demonstrate that compared with the conventional BP decoder, the BLER of the proposed bit-flip decoder can achieve significant signal-to-noise ratio (SNR) gain which is comparable to that of a cyclic redundancy check-aided SC list decoder with a moderate list size. In addition, the decoding latency of the proposed BP bit-flip decoder is only slightly higher than that of the conventional BP decoder in the medium and high SNR regions.

Improved reconciliation with polar codes in quantum key distribution

Quantum key distribution (QKD) is a cryptographic system that generates an information-theoretically secure key shared by two legitimate parties. QKD consists of two parts: quantum and classical. The latter is referred to as classical post-processing (CPP). Information reconciliation is a part of CPP in which parties are given correlated variables and attempt to eliminate the discrepancies between them while disclosing a minimum amount of information. The elegant reconciliation protocol known as \emph{Cascade} was developed specifically for QKD in 1992 and has become the de-facto standard for all QKD implementations. However, the protocol is highly interactive. Thus, other protocols based on linear block codes such as Hamming codes, low-density parity-check (LDPC) codes, and polar codes have been researched. In particular, reconciliation using LDPC codes has been mainly studied because of its outstanding performance. Nevertheless, with small block size, the bit error rate performance of polar codes under successive-cancellation list (SCL) decoding with a cyclic redundancy check (CRC) is comparable to state-of-the-art turbo and LDPC codes. In this study, we demonstrate the use of polar codes to improve the performance of information reconciliation in a QKD system with small block size. The best decoder for polar codes, a CRC-aided SCL decoder, requires CRC-precoded messages. However, messages that are sifted keys in QKD are obtained arbitrarily as a result of a characteristic of the QKD protocol and cannot be CRC-precoded. We propose a method that allows arbitrarily obtained sifted keys to be CRC precoded by introducing a virtual string. Thus the best decoder can be used for reconciliation using polar codes and improves the efficiency of the protocol.