Performance Analysis of CRC-Polar Concatenated Codes

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.

Due to the unsatisfactory block error rate (BLER) performance of short or moderate block , the further development of polar codes is limited. In this paper, the parity-check codes concatenated cyclic redundancy check aided successive cancellation list (CA-SCL) decoding algorithms are studied. Then, an efficient concatenated method, double-parity-check CA-SCL (DP-CA-SCL) polar codes, is proposed according to exiting parity-check decoding algorithms. Not only can this new method improve the BLER performance in short or moderate block, but also it can guarantee the reduction of complexity relative to original CA-SCL method. The simulations in additive white Gaussian noise channel with binary phase shift keying modulation reveal that the DP-CA-SCL polar codes can provide significant performance gains. Comparing to the original CA-SCL decoding, there are 0.5 dB performance gains using proposed method at BLER of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-3</sup> with (128, 64) polar codes. For another moderate block (256, 128) polar codes, it can also guarantee performance gains at least 0.18 dB.

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.

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.

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.

In this paper, we propose a concatenated code by combining a Recursive Systematic Convolutional (RSC) code with a Low-Density Parity-Check (LDPC) code. The proposed concatenated code is encoded in parallel by an RSC encoder and an LDPC encoder without interleavers between them. When decoding the proposed concatenated code, the information bits are decoded by two decoding algorithms, and soft information is exchanged between the RSC decoder and the LDPC decoder. This allows for the elimination of wrong codewords output by each decoder. The use of an LDPC code allows us to halt the decoding process when the valid codeword is obtained. We evaluate the error rate performance of the proposed concatenated code by computer simulations. We show that the proposed concatenated code does not have a high error floor, and achieves better BLock Error Rate (BLER) performance than either conventional LDPC codes or conventional turbo codes in the high Eb/No region. We also show that since the proposed concatenated code can correct wrong codewords output by each decoder, it achieves better BLER than the regular LDPC code at the same Bit Error Rate (BER).

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.

Turbo Codes and Tail Biting Convolutional Codes are used for channel coding in LTE, but these codes are not pertinent in 5G. For Channel coding, advanced error correction techniques are required for the 5G. Polar codes are adopted for control information and LDPC codes are required for Data information for the 5G. Various coding techniques such as LDPC codes, Quasi Cyclic (QC)-LDPC codes, Non-Binary (NB)-LDPC codes, Spatially Coupled (SC)-LDPC codes, Irregular Repeat Accumulate (IRA) codes, Polar codes, Distributed CRC Aided (DCA) codes, and Fountain codes are illustrated. For Decoding Min-Sum Algorithm and Belief Propagation are used for LDPC codes. Min-Sum, OMS, NMS, BP methods are applied in QC-LDPC codes. SC, SCL, FSCL, CASCL, BP algorithms implemented for Polar codes. Simulation of QC-LDPC codes, Polar codes, and LDPC codes are generated by using MATLAB Software. The performance of all channel coding techniques with consideration to BER and BLER is mentioned.

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, we study the block error rate (BLER) performance of spatially coupled low-density parity-check (SC-LDPC) codes using a sliding window decoder suited for streaming applications. Previous studies of SC-LDPC have focused on the bit error rate (BER) performance or the frame error rate (FER) performance over the entire length of the code. Here, we consider protograph-based constructions of SC-LDPC codes in which a window decoder continuously outputs blocks in a streaming fashion, and we examine the BLER associated with these blocks. We begin by examining the effect of protograph design on the streaming BLER by varying the block size and the coupling width in such a way that the overall constraint length of the SC-LDPC code remains constant. Next, we investigate the BLER scaling behavior with block size and coupling width. Lastly, we consider the effect of employing an outer code to protect blocks, so that small numbers of residual errors can be corrected by the outer code. Simulation results for the additive white Gaussian noise channel (AWGNC) are included and comparisons are made to LDPC block codes (LDPC-BCs).

In the case of short code, polar code demonstrates superior performance compared with Low Density Parity Check (LDPC) code. Particularly in low coding rate scenario, the complexity of Polar code encoding and decoding is significantly lower than that of LDPC code. The technological advantage of Polar code makes it of great value in special application scenarios such as satellite communication and ocean communication. The polar coding scheme can significantly improve the transmission reliability of the communication system, because the satellite spread spectrum communication system generally adopts short code design.In the research of decoding algorithm, the original Serial Cancellation (SC) decoding algorithm has simple rules, but the decoding effect is not ideal under the limited code length, and the serial decoding mechanism leads to large decoding delay. The Successive Cancellation List (SCL) decoding algorithm improves decoding performance by increasing the number of decoding search path. However, maintaining multiple decoding paths can multiply the decoding complexity. Fast node-based decoding algorithm aims to accelerate decoding and reduce latency by defining and simplifying the decoding process for specific nodes, yet they do not optimize the decoding paths. In view of the above problems, this paper proposes a Simplified Parity Check-based Cyclic Redundancy Check (CRC)-Aided Successive Cancellation List (S-PC-CASCL) decoding algorithm. Simulation results demonstrate that the proposed S-PC-CASCL maintains excellent error correction performance while reducing the average list size by up to 16 % under low Signal Noise Ratio (SNR) conditions, thereby lowering computational complexity and decoding latency compared with other widely used polar decoding methods.

In this paper, a minimum mean squared error (MMSE) based turbo equalization scheme in delay-Doppler domain for a coded orthogonal time frequency space (OTFS) system is provided as a low complexity version requiring only a single matrix inversion. Also, we derived extrinsic log likelihood ratios (LLRs) for the turbo equalization, exploiting a priori information fed from a channel decoder, under the assumption that the equalized data symbol follows Gaussian distribution. We rigorously conducted link level evaluation to compare coded orthogonal frequency division multiplexing (OFDM) and OTFS over time-varying multipath channels (based on the channel model in 3GPP 5G New Radio) in 30 GHz band, which may cause severe inter-symbol interference (ISI) and/or inter-carrier interference (ICI) depending on numerology setup. The link level simulation verifies that OTFS is superior to OFDM in block error rate (BLER) performance, each being configured with the best numerology, over all the channel scenarios considered. OTFS shows better BLER performance with smaller subcarrier spacing (SCS) while OFDM has an intermediate SCS for the best BLER due to trade-off between ISI and ICI. Also, we measured normalized mean squared error (NMSE) to see how ISI and ICI affect the performance with respect to numerology, which is used to conjecture why the BLER performance has such peculiarity.

Polar codes [1] are a class of provably capacity-achieving error correcting codes (ECC) for memoryless symmetric channels with low-complexity code structures; they have been shown to enjoy better decoding performance over LDPC and Turbo codes under certain scenarios. In this work, we study the application of polar codes for NAND-based SSDs, and mainly focus on the joint source channel coding (JSCC) aspect of polar codes for flash data protection. In a practical SSD system, in addition to the host data, both flash translation layer (FTL) and host side will write on NAND a small amount of non-host data (a.k.a. metadata) for system managing purposes. We demonstrate that the metadata information can be readily and efficiently utilized by polar codes, whereas such property is lacking in LDPC-based systems. With the help of the side information provided by metadata, the decoding capability of polar codes can be further enhanced with little or no modifications on the decoder side. Simulation results verify the performance improvement of metadata-based JSCC scheme, and indicate that polar codes have competitive advantages over LDPC codes on flash memories.

Recently, the importance of channel coding has become prominent for next-generation 5G wireless communication networks. Channel coding is becoming an active area of research and scholars are aiming to improve their knowledge about channel coding schemes which will be playing a crucial part in high-performance 5G networks. polar codes are the class of channel coding techniques that have been standardized in 5G. Moreover, it is a fact that activity tool-based learning raises the learning efficiency and generates intrinsic motivation for learning. Therefore, the main motivation for designing this paper is to develop a Graphical User Interface (GUI) based learning tool to help students easily learn channel coding techniques. In our teaching curriculum, we have used figure-based interactive representations to teach the basic concepts of polar codes efficiently. This learning tool provides users the option to learn about the encoding technique and two different decoding techniques named as Successive Cancellation (SC) and Successive Cancellation List (SCL) for different lengths of user-defined data bits. This complete model has been designed in Python.

This paper presents a general decoder architecture for 5G LDPC and polar codes. To achieve the hardware reuse maximization, both LDPC and polar codes are decoded by belief propagation (BP) algorithm on sparse bipartite graphs. However, by counting the number of short cycles on the polar bipartite graphs, we observe that numerous short cycles cause the poor performance than conventional decoding algorithms, like successive cancellation (SC) and BP decoding. To reduce the impact of short cycles, a LDPC-like BP list (BPL) decoding algorithm is proposed for polar codes, which is applied on L different polar bipartite graphs constructed by different permuted polar factor graphs. Finally, the simulation shows that the proposed general decoder for polar codes can obtain beyond SC and BP error performance with lower decoding latency and computational complexity, and even near successive cancellation list (SCL) performance in high signal-noise ratios (SNRs).