Sphere decoder for polar codes concatenated with cyclic redundancy check

Polar codes are the first provable capacity-achieving channel codes. Despite the splendid performance of long Polar codes, short Polar codes have relatively poor performance compared with other modern channel coding schemes (e.g., Turbo codes and LDPC). In this paper, we explore some practical methods to improve the performance of Polar codes with short to moderate codeword lengths. First, we use Reed Solomon (RS) codes as outer codes. With a specific interleaving strategy, we can concatenate multiple RS codes with one frame of Polar codes. Combining a strategy of allocating unequal RS code rates with the concatenation, different levels of protection are assigned based on the error pattern of successive cancellation list (SCL) decoders. Thus, the finite length performance will certainly be enhanced for this encoding scheme. Meanwhile, the memory size that the original SCL decoding procedure requires is reduced, and the increment of overall decoding complexity is small. Additionally, we propose an intra-frame interleaver to further enhance the performance by dispersing errors. Finally, we designed a list decoding scheme for the proposed multi-RS concatenated Polar codes. Depending on the soft information generated by an SCL decoder, we calculated the reliability of each RS symbol and conducted soft RS decoding. So, the overall performance was enhanced under this joint decoding strategy. Simulation results indicate that the bit error rate (BER) performance of short Polar codes can be well improved.

Cyclic redundancy check (CRC)-aided successive cancellation list (CA-SCL) decoding algorithm has brought good error performance. Recently, some researchers utilized multiple CRC codes to reduce decoding complexity of CA-SCL decoding algorithm. However, based on CA-SCL decoding frame, we not only reduce decoding complexity but also improve error performance. We divide polar codeword into independent sub-blocks during the process of encoding. Meanwhile, locations of multiple CRC are optimized according to a value of each average Bhattacharyya parameter. Finally, for CA-SCL decoding, results indicate that our constructed polar codes show an admirable error performance and low decoding complexity than that of equal sub-block codewords.

Polar codes with dynamically frozen bits are combined with probabilistic amplitude shaping (PAS). At the transmitter, constant composition distribution matching (CCDM) is used to match uniformly distributed input bits to symbols having a desired amplitude distribution. At the receiver, a type check exploits the constant composition properties of CCDM in successive cancellation list (SCL) decoding. For 64 amplitude shift keying (ASK) symbols and a spectral efficiency of 1.5 bits per ASK symbol, polar coded PAS with outer cyclic redundancy check (CRC) codes and SCL decoding with type check outperforms PAS with state-of-the-art LDPC codes by 2dB and performs within 0.8dB of Shannon's sphere packing bound (SPB).

We explore the relationship between polar and RM codes and we describe a coding scheme which improves upon the performance of the standard polar code at practical block lengths. Our starting point is the experimental observation that RM codes have a smaller error probability than polar codes under MAP decoding. This motivates us to introduce a family of codes that "interpolates" between RM and polar codes, call this family C-inter = {C-alpha : alpha is an element of [0, 1]}, where C-alpha vertical bar(alpha=1) is the original polar code, and C-alpha vertical bar(alpha=0) is an RM code. Based on numerical observations, we remark that the error probability under MAP decoding is an increasing function of a. MAP decoding has in general exponential complexity, but empirically the performance of polar codes at finite block lengths is boosted by moving along the family C-inter even under low-complexity decoding schemes such as, for instance, belief propagation or successive cancellation list decoder. We demonstrate the performance gain via numerical simulations for transmission over the erasure channel as well as the Gaussian channel.

This work presents an integrated framework that combines polar coding, spatial modulation, and symbol‐level guessing random additive noise decoding (symbol‐level GRAND) aided detection to enhance the reliability and latency performance of CRC‐aided polar codes in 6G ultrareliable low‐latency communication (URLLC) scenarios. The proposed polar‐coded spatial modulation (PCSM) scheme utilizes (CA‐polar) codes to encode information bits transmitted as antenna index and symbol bits over a MIMO Rayleigh fading channel. The receiver employs a two‐stage detection scheme comprising initial spatial signal processing for antenna and symbol detection, followed by symbol refinement through symbol‐level GRAND with final error correction achieved with CRC‐aided successive cancellation list (CA‐SCL) decoding. A sequential detection and decoding strategy is adopted, in which symbol‐level GRAND refines the detected bits before CA‐SCL polar decoding, reducing error propagation into the polar decoder and thereby lowering the overall decoding complexity. Performance of the scheme is evaluated for polar codeword length of 128 bits for modulation orders of 16‐QAM and 64‐QAM, highlighting the scheme's parametric flexibility under short codeword constraints. Simulations carried out in MATLAB demonstrate that error correction with symbol‐level GRAND‐aided detection achieves an improved bit error rate (BER) and a lower system complexity facilitating real‐time hardware realization.

Polar codes are used for image transmission due to unequal error protection. Meanwhile, a low-complexity polar code decoder helps to reduce the energy loss during polar code decoding. Therefore, this paper proposes a new decoder with low complexity, called adaptive distributed cyclic-redundancy-checkaided successive cancellation list decoder with flip operations (ADD-SCLF). By combining the adaptive scheme and the distributed cyclic redundancy check (CRC) bit scheme, the decoder can find the correct path with lower average computational complexity and end the failed decoding process earlier simultaneously. Simulation results show that our ADD-SCLF decoder can reduce the computational complexity by up to 17.16% compared to the current adaptive ordered shifted-pruning list (ADOSPL) decoder.

Recently, an intensive discussion on candidate channel coding schemes for the fifth-generation (5G) mobile communication systems has been carried out both in academia and industry. We added to this discussion by providing a comparison between two capacity-achieving channel codes: Low-density Parity-Check (LDPC) codes and Polar codes. The considered codes have been selected as the standard for 5G New Radio (NR) by the 3rd Generation Partnership Project (3GPP). We investigate these codes in terms of Frame Error Rate (FER) on small and moderate length message transmission in the context of Massive Machine type communication (mMTC). Multiple decoding implementation schemes are considered for both codes and compare their error correction performance over an Additive White Gaussian Noise (AWGN) channel. The simulation results reveal that polar codes under the Cyclic redundancy code (CRC) aided Successive Cancellation list (CA-SCL) decoding performs better as compared to other decoding patterns in terms of reliability. Furthermore, we analyze the impact of CRC length on the decoding performance of CA-SCL.

In this paper, a new list decoding algorithm for tail-biting convolutional codes (TBCCs) with a cyclic redundancy check (CRC) is proposed, where the CRC is considered as a concatenated outer code. The main idea of the proposed algorithm is to modify the list decoding procedure of the TBCC by using the CRC. Two algorithms are proposed for the list decoding of the TBCC with the CRC. The first proposed algorithm is a new initial state estimating algorithm using re-encoded CRC bits and having the low computational complexity. The other proposed algorithm is a modified list Viterbi algorithm, where trellis paths are fixed by re-encoded CRC bits and some CRC bits are used for the error correction. For the TBCC concatenated with the CRC code defined in the long-term evolution standard, the proposed decoding scheme by partially using CRC bits outperforms the conventional list decoding algorithms for the list size $L=4$ even though the proposed algorithm has the lower decoding complexity.

This paper presents a novel methodology for designing the reliability sequence π for polarization-adjusted convolutional (PAC) and polar codes. Given a sequence π of length N, a family of nested information sets is constructed for N codes. The design objective is to minimize the average of the minimum signal-to-noise ratios (SNRs) required to achieve a target block error rate (BLER) across these nested codes. To reduce the search space, the N bit positions are partitioned into M components of length N/M, with bit positions within each component following a predefined reliability order. The design problem is then formulated as a Markov decision process (MDP) and solved using a deep Q-network (DQN) with an experience replay buffer. Simulation results under CRC-aided successive cancellation list (CA-SCL) decoding for nested polar codes show that the proposed sequence improves SNR performance over existing nested polar codes, including those used in 5G. Additionally, nested PAC codes under successive cancellation list (SCL) decoding with the proposed π outperform nested PAC and polar codes based on the known reliability sequences, under SCL or CA-SCL decoding. Moreover, individual PAC codes extracted from the nested family perform competitively with the best-known PAC/polar codes designed for specific rates. The proposed method is applicable to various precoded polar code constructions and decoding algorithms. The observed performance improvements, particularly in PAC codes, suggest that this approach is a promising candidate for future communication systems, including 6G.

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 codes are of great interest, since they are the first provably capacity-achieving forward error correction codes. To improve throughput and to reduce decoding latency of polar decoders, maximum likelihood (ML) decoding units are used by successive cancellation list (SCL) decoders as well as SC decoders. This paper proposes an approximate ML (AML) decoding unit for SCL decoders first. In particular, we investigate the distribution of frozen bits of polar codes designed for both the binary erasure and additive white Gaussian noise channels, and take advantage of the distribution to reduce the complexity of the AML decoding unit, improving the throughput-area efficiency of the SCL decoders. Furthermore, a multimode (MM) SCL decoder with variable list sizes and parallelism is proposed. If high throughput or small latency is required, the decoder decodes multiple received words in parallel with a small list size. However, if error performance is of higher priority, the MM-SCL decoder switches to a serial mode with a bigger list size. Therefore, the MM-SCL decoder provides a flexible tradeoff between latency, throughput, and error performance at the expense of small overhead. Hardware implementation and synthesis results show that our polar decoders not only have a better throughput-area efficiency but also easily adapt to different communication channels and applications.

Cyclic redundancy check (CRC) bits that are conventionally used for error detection have recently found a new application in universal mobile telecommunications system standard for message length detection of variable-length message communications. It was anticipated that the CRC bits, when they are coworked with the inner convolutional code, can be used to detect the receiver-unaware of the message length-without much degradation in their error detection capability. This is unfortunately not true when the offset or difference between the wrong detected length and the true length is small. Two improvements, i.e., the DoCoMo's reverse CRC method and the flip CRC method, were accordingly proposed. In this paper, we revisited the flip CRC modification by considering the impact of joint decoding of the CRC code and the convolutional code. By generalizing the condition for the selection of the flip polynomials, we found that under error-free transmission, the range of the length offsets, at which the false length probability conditioning on the true message length can be made exactly zero (and hence, is minimized), can be extended from to , where and are, respectively, the number of the CRC bits and the memory order of the convolutional code. In addition, an upper bound and a lower bound for the overall false length probability with respect to a uniform pick of the true message length over a candidate message length set are derived. It is then confirmed numerically that the two bounds almost coincide for moderate value. Simulations show that the false length probability obtained analytically under error-free transmission assumption only mildly degrades for moderate-to-high SNRs. Interestingly, we also found that the system block error rate of the flip CRC method can be well approximated by the performance curve of the adopted convolutional code up to a certain SNR, and approach an error floor determined well by the previously derived false length probability bounds beyond this SNR, thereby facilitating the selection of the system parameters, such as the number of CRC bits and the memory order of the convolutional code.

Polar codes were shown to be weak at finite block lengths even under maximum-likelihood (ML) decoding. To enhance polar codes, CRC-aided polar codes were proposed. It has been shown that CRC-aided polar codes under the successive cancellation list (SCL) decoder outperform LDPC codes and turbo codes. Recently, interleaved polar (i-polar) codes were proposed. By using a new bit channel selection algorithm, i-polar codes without CRC were shown to outperform the 5G LDPC codes. However, most research on polar codes focuses on the binary case. In this paper, a new family of nonbinary i-polar codes with random constellation mapping is proposed. A recursive form for calculating the average weight enumerating function (WEF) of the nonbinary i-polar code is derived. With the random constellation mapping, the upper bound on the BLER under ML decoding depends on the WEF of the nonbinary i-polar code. Then a new channel selection algorithm is proposed based on the polarization effect and the BLER upper bound. The algorithm generates a sequence, termed the i-polar sequence (IPS). Simulation results show that the nonbinary i-polar codes with the IPS outperform the 5G LDPC codes.

Irregular LDPC codes can achieve better error rate performance than regular LDPC codes. However, irregular LDPC codes have higher error floors than regular LDPC codes. The Ordered Statistic Decoding (OSD) algorithm achieves approximate Maximum Likelihood (ML) decoding. ML decoding is effective to lower error floors. However, the OSD estimates satisfy the parity check equation of the LDPC code even the estimates are wrong. Hybrid decoder combining LLR-BP decoding algorithm and the OSD algorithm cannot also lower error floors, because wrong estimates also satisfy the LDPC parity check equation. We proposed the concatenated code constructed with an inner irregular LDPC code and an outer Cyclic Redundancy Check (CRC). Owing to CRC, we can detect wrong codewords from OSD estimates. Our CRC-LDPC code with hybrid decoder can lower error floors in an AWGN channel. In wireless communications, we cannot neglect the effects of the channel. The OSD algorithm needs the ordering of each bit based on the reliability. The Channel State Information (CSI) is used for deciding reliability of each bit. In this paper, we evaluate the Block Error Rate (BLER) of the CRC-LDPC code with hybrid decoder in a fast fading channel with perfect and imperfect CSIs where 'imperfect CSI' means that the distribution of channel and those statistical average of the fading amplitudes are known at the receiver. By computer simulation, we show that the CRC-LDPC code with hybrid decoder can lower error floors than the conventional LDPC code with hybrid decoder in the fast fading channel with perfect and imperfect CSIs. We also show that combining error detection with the OSD algorithm is effective not only for lowering the error floor but also for reducing computational complexity of the OSD algorithm.

Error correction performance of finite-length polar codes has been a major concern in the past decade. This Letter presents a method for improving the error correction performance of cyclic redundancy check (CRC)-aided successive cancellation list decoding for polar codes of medium block-lengths with high code rates. In the proposed decoding scheme, a one-time repetition of least reliable bit-channels along with partial protection of the message is employed to correct the error(s) when decoding fails. The results show an improvement of nearly 0.2 dB in error correction performance.