Short Polar Codes Research Articles

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.

Low complexity symmetric-coded based sphere decoding for low-rate polar codes

The sphere decoding (SD) algorithm can provide (sub)optimal solutions with reduced computational complexity of maximum likelihood (ML) detection for multi-input multi-output (MIMO) communication systems. In this paper, we propose a novel low complexity symmetric-coded based SD algorithm for short polar codes with low rate. At the encoding stage, the first N/2 sub-channels transmit the frozen bits, while the information bits are selected from the latter N/2 sub-channels. Two symmetric codes are generated due to the mathematical structure of the generator matrix, which is well conditioned to the SD search. At the decoding stage, the presented SD algorithm computes the Euclidean distance value by the combined signals to estimate the latter N/2 input bits. Furthermore, the backtrack operation starts from the earlier (N/2+1)-th bit, which can significantly reduce the average visited nodes (AVN). Simulation results show that, compared to the original SD algorithm, the presented variant of the SD algorithm can reduce the AVN to 0.9% for the polar code P(64, 14) at SNR = 1 dB with a performance loss within 0.2 dB. The presented SD algorithm may find applications in MIMO systems where the complexity of the standar ML detection increases exponentially with the transmitting antennas.

CRC-Aided Adaptive BP Decoding of PAC Codes.

Although long polar codes with successive cancellation decoding can asymptotically achieve channel capacity, the performance of short blocklength polar codes is far from optimal. Recently, Arıkan proposed employing a convolutional pre-transformation before the polarization network, called polarization-adjusted convolutional (PAC) codes. In this paper, we focus on improving the performance of short PAC codes concatenated with a cyclic redundancy check (CRC) outer code, CRC-PAC codes, since error detection capability is essential in practical applications, such as the polar coding scheme for the control channel. We propose an enhanced adaptive belief propagation (ABP) decoding algorithm with the assistance of CRC bits for PAC codes. We also derive joint parity-check matrices of CRC-PAC codes suitable for iterative BP decoding. The proposed CRC-aided ABP (CA-ABP) decoding can effectively improve error performance when partial CRC bits are used in the decoding. Meanwhile, the error detection ability can still be guaranteed by the remaining CRC bits and adaptive decoding parameters. Moreover, compared with the conventional CRC-aided list (CA-List) decoding, our proposed scheme can significantly reduce computational complexity, to achieve a better trade-off between the performance and complexity for short PAC codes.

A hybrid Sphere Decoding for short polar codes using variable step size

In this paper, a hybrid stack Sphere Decoding (SD) for short polar codes based on radius extension and a novel path metric (PM) is proposed. The proposed SD scheme utilizes variable step size obtained from a novel PM, which is a modified PM from a previous work. The proposed method continuously increases the corresponding variable radius until a valid polar codeword is found. Moreover, by introducing a limited size stack, the proposed scheme can avoid node revisiting and achieve a trade-off between performance and complexity. Simulation results show that for short polar codes, compared with existing SD schemes that achieve maximum likelihood (ML) performance, the proposed scheme achieves more complexity reduction without compromising the ML optimality.

DLSTM-Based Successive Cancellation Flipping Decoder for Short Polar Codes.

Polar code has been adopted as the control channel coding scheme for the fifth generation (5G), and the performance of short polar codes is receiving intensive attention. The successive cancellation flipping (SC flipping) algorithm suffers a significant performance loss in short block lengths. To address this issue, we propose a double long short-term memory (DLSTM) neural network to locate the first error bit. To enhance the prediction accuracy of the DLSTM network, all frozen bits are clipped in the output layer. Then, Gaussian approximation is applied to measure the channel reliability and rank the flipping set to choose the least reliable position for multi-bit flipping. To be robust under different codewords, padding and masking strategies aid the network architecture to be compatible with multiple block lengths. Numerical results indicate that the error-correction performance of the proposed algorithm is competitive with that of the CA-SCL algorithm. It has better performance than the machine learning-based multi-bit flipping SC (ML-MSCF) decoder and the dynamic SC flipping (DSCF) decoder for short polar codes.

Fast and efficient FPGA implementation of Polar Codes and SoC test bench

In this paper we describe a novel and efficient System on Chip Field Programmable Gate Array (SoC FPGA) implementation and test bench for short Polar Codes on an Intel DE10-Standard Development Kit. Encoder and decoder are synthesized on the FPGA fabric and the whole functionality of a complete test bench is developed with an embedded ARM-based hard processor system. A Very High Speed Integrated Circuit (VHSIC) Hardware Description Language (VHDL) parametric design allows synthesis for different code lengths and is suitable for rate-adaptive decoding schemes. We implement fully-unrolled encoder and decoder architectures to achieve high troughputs and lower energy requirements, and achieve an 11% higher throughput than a reference implementation, for short Polar Codes. A novel Merged Processing Element (MPE) is optimized to be used with Sign–Magnitude LLR (SM LLR) discrete representations and pre-computing results in the decoder to reduce latency and resource consumption in comparison to reference designs. A simplified version of this MPE is also implemented, trading higher latencies for lower resource requirements. The SoC test bench design allows single-board automated testing and is also suitable for other error-correcting schemes. Error-correcting performance is evaluated for different combinations of integer and decimal part bits in LLR quantified representations. Also a new simplified non-statistical LLR metric was tested with promising results.

Design of Short Polar Codes for SCL Decoding

The problem of designing polar-like codes for successive cancellation list (SCL) decoding algorithm is considered. A novel code design algorithm aimed to minimize both successive cancellation (SC) and maximum-likelihood (ML) decoding error probabilities is introduced. The algorithm performs optimization of a precoding matrix for the polarization transformation matrix to guarantee preselected minimum distance and the lowest possible SC decoding error probability to the resulting code. Constructed precoded polar codes are decoded as polar codes with dynamic frozen bits. Numerical results show that the proposed codes of lengths 32 and 64 significantly reduce the frame error rate (FER) compared to the original polar codes under the SCL decoding starting with the list size 4. The gain increases with the list size. The proposed codes have a much lower SC decoding error probability than extended Bose-Chaudhuri-Hocquenghem (e-BCH) codes and their polar subcodes without sacrificing the ML performance. In case of the code length 64, the constructed precoded polar codes demonstrate a FER reduction compared to the polar subcodes of e-BCH codes under the SCL decoding with the list size up to 16 and demonstrate a comparable FER starting with the list size 32.

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.

A Modified Approach to Punctured Product Polar Codes

A high-performance polar code introduced as a product polar code can be constructed by concatenating two short length polar codes. The punctured structure of this code was achieved by puncturing one of its constituent codes. The constructed polar code provided better performance than a single polar code in the error floor region. However, its performance in the waterfall region was not as good as that of single polar codes. This paper proposes a new puncturing algorithm for product codes constructed by two identical polar codes. Puncturing is conducted on both constituent codes, to ensure that the new code outperforms the previously punctured product polar and single polar codes. This is evident in both waterfall and error floor regions.

A Low-Complexity Ordered Statistics Decoding Algorithm for Short Polar Codes

In this paper, we propose a low-complexity ordered statistics decoding (OSD) algorithm called threshold-based OSD (TH-OSD) that uses a threshold on the discrepancy of the candidate codewords to speed up the decoding of short polar codes. To determine the threshold, we use the probability distribution of the discrepancy value of the maximal likelihood codeword with a predefined parameter controlling the trade-off between the error correction performance and the decoding complexity. We also derive an upper-bound of the word error rate (WER) for the proposed algorithm. The complexity analysis shows that our algorithm is faster than the conventional successive cancellation (SC) decoding algorithm in mid-to-high signal-to-noise ratio (SNR) situations and much faster than the SC list (SCL) decoding algorithm. Our addition of a list approach to our proposed algorithm further narrows the error correction performance gap between our TH-OSD and OSD. Our simulation results show that, with appropriate thresholds, our proposed algorithm achieves performance close to OSD’s while testing significantly fewer codewords than OSD, especially with low SNR values. Even a small list is sufficient for TH-OSD to match OSD’s error rate in short-code scenarios. The algorithm can be easily extended to longer code lengths.

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.

Mapping Design for $2^{M}$ -Ary Bit-Interleaved Polar Coded Modulation

This paper proposes a mapping design for bit-interleaved polar coded modulation (BIPCM) systems with belief propagation (BP) decoding. We first introduce a two-layer bipartite graph to represent BIPCM, where a new mapping graph linking polar graph to modulator is added to the conventional factor graph. Then, a mapping design is proposed and the design paradigm is to separate sub-channels with lower reliability to different stopping trees of polar codes, aiming to make sure that each stopping tree receives reliable extrinsic information from demodulator. The proposed mapping algorithm is employed for BIPCM with traditional polar codes over 16-quadrature amplitude modulation (QAM) and 256-QAM. Numerical results show that our scheme can improve the error-correcting performance compared to the conventional scheme with a random mapping. Furthermore, to meet code-length requirement of different modulation orders, we propose an efficient method to construct flexible-length polar code (FLPC) by coupling several short length polar codes with a repeat-accumulate (RA) code. Also, the proposed FLPC is employed in the BIPCM system, with the designed mapping algorithm, simulation result also reveals that the block error rate performance of proposed BIPCM scheme with BP decoding outperforms the one with successive cancellation decoding by providing a gain up to 1 dB.

High Performance Short Polar Codes: A Concatenation Scheme Using Spinal Codes as the Outer Code

An innovative scheme for concatenating the polar codes with the recently proposed spinal codes is presented in this paper. The primary objective is to improve the performance of the polar codes in the finite-length regime. One effective used way is to concatenate the polar codes with short outer codes. Essentially, the short outer codes that have as high performance as possible is required for concatenation. To this end, the newly invented spinal codes are used as the outer codes, which are characterized by its great error correcting capability even when the message length is short. First, the proposed codes, named spinal-polar, is implemented through an interleaved concatenation scheme. A joint iterative decoding algorithm is proposed accordingly, and the decoding complexity of the algorithm is also analyzed. Second, in order to reduce the decoding complexity of the full concatenation while maintaining the BER performance, the partial concatenation scheme is presented. Finally, we propose a rate allocation algorithm to further improve the BER performance of the designed concatenated codes. Extensive simulations results indicate that the proposed spinal-polar codes have significant performance improvement over the stand-alone polar codes, and the other improvement strategies mentioned earlier have also been proved to be effective.

An Enhanced Probabilistic Computation Method for the Weight Distribution of Polar Codes

Owning to the explicit structure, Hamming weight distribution (HWD) of polar codes with large length can be recursively computed with that of the short one. In this letter, an enhanced probabilistic computation method is proposed to estimate the HWD of polar codes. Only the uniform distribution assumption for the 1 s in the codeword is used and the exhaustively searched HWD for very short polar code are combined to improve the accuracy of the proposed method. Meanwhile, some characteristics for the HWD of polar codes are discussed and used to avoid the unnecessary calculations. Simulation results show that the estimated HWD matches the exact one well for rate larger than 0.5. With the obtained HWD, the asymptotic performance of polar codes can be accurately estimated in almost the whole rate range.

Improved Belief Propagation Decoding Algorithm for Short Polar Codes

In this paper, we discuss the belief propagation (BP) decoding of polar codes. The performance of polar codes for short lengths is not satisfactory. Therefore, motivated by this we propose a novel technique to improve the performance of polar codes under BP decoding. In order to enhance the reliability of the variable nodes’ propagated messages in BP decoding, the messages are multiplied by previous messages of variable nodes. It is also shown that the BP decoding can be performed on polar codes if the factor graph is constructed according to the parity check matrix of polar codes. Simulation results in binary-input additive white Gaussian noise channel show that using the proposed method a gain of 1–2 dB can be achieved. We also show that the complexity of proposed decoder is same as the complexity of standard BP decoder. Furthermore, we show that the proposed decoder requires about 10–25 iterations less than the standard BP decoder.

Ordered Statistic Decoding for Short Polar Codes

In this letter, the ordered statistic decoding (OSD) is considered to decode short polar codes. Since the number of tested codewords is high in the original OSD, a threshold-based OSD is proposed to reduce the complexity. The analysis shows that the threshold affects the OSD in two ways. One is the error floor in high signal-to-noise ratio (SNR) region while the other is the reduction of the number of tested codewords. The estimation on the BLER performance in high SNR region and the average number of tested codewords are discussed. After that, CRC-aided OSD is proposed to further improve the performance of polar codes. Simulation results show that a well-designed threshold-based OSD can reduce the complexity without significant performance loss. And a higher order OSD decoder is needed if OSD is used to decoding moderate or long polar codes. It is also illustrated that the CRC-aided OSD outperforms the CRC-aided list decoding when the code rate is high. When code rate is low, the result inverses.

An Efficient List Decoder Architecture for Polar Codes

Long polar codes can achieve the symmetric capacity of arbitrary binary-input discrete memoryless channels under a low complexity successive cancelation (SC) decoding algorithm. However, for polar codes with short and moderate code length, the decoding performance of the SC algorithm is inferior. The cyclic redundancy check (CRC) aided successive cancelation list (SCL) decoding algorithm has better error performance than the SC algorithm for short or moderate polar codes. In this paper, we propose an efficient list decoder architecture for the CRC aided SCL algorithm, based on both algorithmic reformulations and architectural techniques. In particular, an area efficient message memory architecture is proposed to reduce the area of the proposed decoder architecture. An efficient path pruning unit suitable for large list size is also proposed. For a polar code of length 1024 and rate $\frac{1}{2}$, when list size $L=2$ and 4, the proposed list decoder architecture is implemented under a TSMC 90nm CMOS technology. Compared with the list decoders in the literature, our decoder achieves 1.33 to 1.96 times hardware efficiency.

Low-Complexity Sphere Decoding of Polar Codes Based on Optimum Path Metric

Sphere decoding (SD) of polar codes is an efficient method to achieve the error performance of maximum likelihood (ML) decoding. But the complexity of the conventional sphere decoder is still high, where the candidates in a target sphere are enumerated and the radius is decreased gradually until no available candidate is in the sphere. In order to reduce the complexity of SD, a stack SD (SSD) algorithm with an efficient enumeration is proposed in this paper. Based on a novel path metric, SSD can effectively narrow the search range when enumerating the candidates within a sphere. The proposed metric follows an exact ML rule and takes the full usage of the whole received sequence. Furthermore, another very simple metric is provided as an approximation of the ML metric in the high signal-to-noise ratio regime. For short polar codes, simulation results over the additive white Gaussian noise channels show that the complexity of SSD based on the proposed metrics is up to 100 times lower than that of the conventional SD.

A parallel decoding algorithm for short polar codes based on error checking and correcting.

We propose a parallel decoding algorithm based on error checking and correcting to improve the performance of the short polar codes. In order to enhance the error-correcting capacity of the decoding algorithm, we first derive the error-checking equations generated on the basis of the frozen nodes, and then we introduce the method to check the errors in the input nodes of the decoder by the solutions of these equations. In order to further correct those checked errors, we adopt the method of modifying the probability messages of the error nodes with constant values according to the maximization principle. Due to the existence of multiple solutions of the error-checking equations, we formulate a CRC-aided optimization problem of finding the optimal solution with three different target functions, so as to improve the accuracy of error checking. Besides, in order to increase the throughput of decoding, we use a parallel method based on the decoding tree to calculate probability messages of all the nodes in the decoder. Numerical results show that the proposed decoding algorithm achieves better performance than that of some existing decoding algorithms with the same code length.