Multi-RS Concatenated Polar Codes with Enhanced Interleaving and List Decoding

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.

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.

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).

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.

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.

This paper investigates properties of polar codes that can be potentially useful in real-world applications. We start with analyzing the performance of finite-length polar codes over the binary erasure channel (BEC), while assuming belief propagation as the decoding method. We provide a stopping set analysis for the factor graph of polar codes, where we find the size of the minimum stopping set. We also find the girth of the graph for polar codes. Our analysis along with bit error rate (BER) simulations demonstrate that finite-length polar codes show superior error floor performance compared to the conventional capacity-approaching coding techniques. In order to take advantage from this property while avoiding the shortcomings of polar codes, we consider the idea of combining polar codes with other coding schemes. We propose a polar code-based concatenated scheme to be used in Optical Transport Networks (OTNs) as a potential real-world application. Comparing against conventional concatenation techniques for OTNs, we show that the proposed scheme outperforms the existing methods by closing the gap to the capacity while avoiding error floor, and maintaining a low complexity at the same time.

A new family of modified polar codes and a new permutation selection scheme for belief propagation list (BPL) decoding are presented. We first propose a new code construction methodology to interpolate between Reed-Muller (RM) codes and polar codes. By taking advantage of an existing partial order on bit-channels whose corresponding indices share the same Hamming weight, we analyze the complexity of the new construction method. It is shown that we need to compute the reliability of roughly a fraction 1/log <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3/2</sup> N of all the bit-channels contained in the subset. Then, we explore a special family of factor-graph layer permutations called stable permutations (SPs) that preserve a specified information set when the corresponding bit permutations are applied to message bit indices. Simulation results show that the error-rate performance of the new family of codes is better than that of 5G polar codes under successive cancellation list (SCL) decoding. In addition, the SP selection scheme stands out as a preferred one for BPL decoding in terms of error-rate performance, while the average number of iterations per belief propagation (BP) decoder on the permuted factor graph is close to that of the original BP decoder when an early stopping condition is applied.

As the first provably capacity-achieving codes, polar codes have been selected as the coding scheme for 5G enhanced mobile broadband (eMBB) control channels. Furthermore, if the bottleneck in decoding complexity is addressed, polar codes are still promising for long code length scenarios in the future with superior error-correcting performance. Considering the demand of cloud radio access network (C-RAN) in 5G system, the low-latency software polar decoder is studied in this paper. Two novel algorithms are proposed to reduce the latency of the successive cancellation list (SCL) decoding, including the segmented list-pruning (SLP) algorithm and the modified distributed sorting (MDS) algorithm. Moreover, the proposed SLP algorithm improves the decoding performance compared to existing segmentation schemes. In terms of the software implementation, our decoder is nearly $20\%$ faster than the state-of-the-art flexible software polar list decoder.

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.

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.

Deploying polar codes in ultra-reliable low-latency communication (URLLC) has received tremendous attention in both academia and industry. However, the state of the art polar codes decoders, e.g., successive cancellation (SC) and successive cancellation list (SCL) decoder, involve strong data dependencies and suffer from huge decoding delay. This contradicts the low-latency requirement in URLLC. To address this issue, this paper appeals to parallel computing and proposes an adaptive ordered statistic decoder (OSD). Capitalizing on a novel codeword searching metric, the proposed decoder proves to be hardware-friendly and can adaptively rule out the unpromising codewords, thus reducing the latency significantly. Numerical results show that the adaptive OSD outperforms the conventional decoders in terms of block error rate (BLER) and decoding latency.

In this paper, polar codes, low-density parity-check (LDPC) codes and convolutional codes in the status of high-performance wireless (WirelessHP) are discussed. Those classic code schemes are compared in terms of their encoding and decoding complexity and the error-correcting performance. Results show that, polar codes with successive cancellation list (SCL) decoding is superior to the LDPC codes and the convolutional codes in the condition of WirelessHP. The result has certain significance for the development of next generation communication technology.

In this letter, we consider the parity-check polar (PC-Polar) code design under belief propagation list (BPL) decoding. First, we employ a Monte-Carlo method with BPL decoding to identify unreliable bits, from which parity-check (PC) bits are selected. Then the partial orders of the inner polar code are leveraged to develop a practical correlation metric for designing PC constraints. We also utilize the PC constraints to propose an improved BPL decoding with bit flipping. Simulations show the designed PC-Polar codes outperform state-of-the-art PC-Polar codes under BPL decoding, and the proposed BPLF decoding approaches successive cancellation list (SCL) decoding performance.

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.

Since the inherent serial nature of successive cancellation list (SCL) decoding results in a long latency, belief propagation (BP) decoding for polar codes has drawn attention for high-throughput applications. However, its error correction performance is inferior to that of SCL decoding. Therefore, the bit-flipping strategy has been recently applied to BP decoding, which can approach the SCL decoding performance through multiple additional decoding attempts. The original BP flip (BPF) decoding suffers from an inaccurate identification of erroneous bits by a fixed flip set (FS), which has been improved by the generalized BPF (GBPF) decoding. In this article, the GBPF decoding is extended to support multiple bits being flipped in one decoding attempt. In addition, for two types of decoding errors: detected errors and undetected errors, we propose two novel methods to more effectively identify erroneous bits. For detected errors, the concept of loop sets is defined and a loop-based identification method is introduced based on the study of error patterns of BP decoding. On the other hand, a method to generate a more accurate fixed FS is proposed for undetected errors, which considers the bit error distribution under BP decoding. Combining the two methods, the GBPF with merged sets (GBPF-MS) decoding can achieve the SCL-8 performance and outperforms the state-of-the-art BPF, BP list, and SC flip (SCF) decoding, for polar codes with length 1024 and information rate 1/2. Implemented by 40nm CMOS technology, the proposed GBPF-MS decoder with ten flips exhibits an average throughput of 4.19 Gbps at 2.5 dB, which is $1.6\times $ and $1.72\times $ faster than the state-of-the-art SCL-4 and SCF decoders, respectively.