Polar Coding with Introducing Half-Frozen Bits

Space-time block code (STBC) based on orthogonal design is one of the space-time coding methods to achieve multi-input multi-output (MIMO) diversity gain. STBC can provide sufficient space diversity gain, but it can not provide coding gain. Polar codes based on channel polarization are the first class of provably capacity-achieving codes with low encoding and decoding complexity. In order to improve the bit error rate (BER) performance of STBC, a concatenated scheme with Polar codes and STBC, named Polar-STBC scheme, is proposed in this paper. The BER performance of Polar-STBC system is discussed by numerical simulation under different polar code length, different polar code rates, and different coding methods of STBC. The results show that the performance of Polar-STBC has a big improvement in comparison with that using STBC only. The coding gain of Polar-STBC system is about 5dB when the BER is 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-3</sup> . Meanwhile, the longer code length and the lower code rate Polar codes have, the better BER performance the Polar-STBC system has. In addition, the different STBC coding methods have influence on the Polar-STBC system. The Polar-STBC system with more receiving antennas has a better BER performance in comparison with those with the same sending antennas and less receiving antennas. The coding gain of Polar-STBC system with 2 receiving antenna is 6.5dB improvement to the system with 1 receiving antenna when BER is 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-3</sup> and the code length is 1024.

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 research using a polar code and without polar codes -based broadband channel that is affected by human blockage using one of the 5G cellular network frequencies at 2.3 GHz, 99 MHz bandwidth, 128 blocks of Fast Fourier Transform (FFT) with Cyclic prefix-Orthogonal Frequency Division Multiplexing ( CP-OFDM) and Binary Shift Keying (BPSK) modulation. The use of high frequencies causes the technology to be sensitive to the surrounding environment and attenuation such as human blockage. The purpose of this research is to determine the performance results and analyze the BER parameters that use polar codes and without polar codes on 5G network broadband channels that are affected by human blockage. Broadband channel modeling on a 5G network is presented in a representative Power Delay Profile (PDP) with the influence of human blockage, which is obtained as many as 41 paths which have multiple delays of 10 ns on each path. This research also uses the scaling method on representative PDP because the use of FFT will produce 128 blocks, and the results of this scaling show that there are 9 lanes with multiple delays of 50 ns. The results of this study are close to the average Bit Error Rate (BER) of 10-4. BER performance without polar code is affected by human blockage requires Signal to Noise (SNR) of 30 dB, for theory BER on BPSK modulation requires SNR of 34.5 dB and BER performance using polar code only requires SNR of 23 dB. These results indicate that using a polar code can reduce or save power usage by 7 dB without a polar codes. Polar codes can minimize errors in the 5G network system, because polar codes are one of the strong codes and are one of the channel coding recommended by ITU to be applied to 5G network systems

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.

Noise is a major obstacle in the development of practical schemes for quantum computation and communication. Similar to the case of classical communication, this noise can be protected against by employing a code, which provides a means for encoding quantum states prior to transmission and allows for errors to be inferred, and hopefully corrected, by a decoder at the receiver. Unfortunately, designing good codes and decoders is typically a difficult problem. This thesis focuses on developing low-complexity heuristic approaches to three such problems: the design of modified belief propagation decoders for quantum low-density parity-check codes, the design of stabilizer codes for asymmetric channels, and the design of codeword stabilized codes. Quantum low-density parity-check codes are stabilizer codes with low-weight generators. Such codes permit low-complexity decoding via the use of belief propagation, which is an iterative message passing algorithm that takes place on a factor graph defined by the code. However, the performance of such a decoder is limited both by code structure and the degenerate nature of quantum errors. To overcome these limitations, at least in part, a number of modifications to belief propagation are developed. Central among these is the augmented decoder, which in the case of a decoding error, iteratively reattempts decoding using modified factor graph. This heuristic modification simply involves the duplication of a randomly selected subset of the graph’s check nodes, which are in one-to-one correspondence with the code’s stabilizer generators. Across a range of codes, it is shown that the decoders developed perform as well as or better than other modified decoders presented in literature. For a number of channels of physical interest, phase-flip errors occur far more frequently than bit-flip errors. When transmitting across these so-called asymmetric channels, the decoding error rate can be minimized by tailoring the code used to the channel. However, assessing the performance of codes on a given channel is made difficult by the #P-completeness of optimal decoding. To address this complexity, it is shown that the decoding error rate can be accurately approximated using only a small fraction of the possible errors caused by the channel. This approximation is then used to identify a number of cyclic stabilizer codes that perform well on two different asymmetric channels. To further build on this, a heuristic is demonstrated for assessing code performance based on the decoding error rate of an associated classical code. The complexity of calculating this classical error rate is relatively low, and it is shown that it can be used as the basis for a hill-climbing search algorithm. Such searches have yielded a large number of highly performant codes satisfying various structure constraints. The family of codeword stabilized codes encompasses both the stabilizer codes as well as many of the best known nonadditive codes. Constructing a standard form codeword stabilized code is a matter of selecting a simple undirected graph and a binary classical code. This makes designing optimal codes difficult as the number of possible graphs grows exponentially with code length, and the clique search required to construct the classical code is NP-hard. To address the exponential growth of the search space, a heuristic is developed for assessing graphs. This heuristic is then employed by a genetic algorithm that also makes use of a novel crossover operation based on spectral bisection, which is show to be superior to more standard crossover operations. With a graph selected, it is demonstrated that the complexity of the clique search required to construct the associated classical code can be mitigated through the use of a heuristic clique finding algorithm. A number of best known codes are presented that have been found using this approach.

Modified Packet Combining (MPC) scheme is an error correction scheme, which can correct double or, higher bit errors. In some earlier work, a scheme known as Packet Reversed Packet Combining (PRPC) scheme that will correct error if occurs at same bit location of erroneous copies, was studied. Both PRPC and MPC scheme are believed to offer higher throughput and better error correction capability at high bit error rate (BER) and large packet size only. Exact analysis was performed to substantiate this claim of higher throughput and better error correction capability in high as well as in low bit error rate (BER) for large and medium packet size. To combat high bit error rate some researches proposed to use Error Correction Code (ECC) in place of Error Detection Code (EDC) in ARQ scheme. But no conclusive result is found to justify the idea in terms of throughput. The current investigation is an attempt to explore the basic gap. In this paper, we investigate the exact analysis of both PRPC and MPC in a Modified way over Conventional PRPC, and report the findings thereof. We propose a combined protocol of Modified PRPC and MPC, and find that it will offer higher throughput and better error correction capability at high and low BER with medium and large packet size as we have used MPC (Modified Packet Combining) scheme with PRPC (Packet Reverse Packet Combining) scheme in a Modified way by combining BEC (Backward Error Correction) with FEC (Forward Error Correction) Codes for Mobile Communication and Networks.

Motivated by the capacity-approaching property of both binary polar codes (PCs) and the multi-level coding (MLC) scheme, we consider the design of modulated PC over additive white Gaussian noise (AWGN) channels. The capability of modulated PC essentially depends on the selection of information bits set from all the virtual bit channels formed by both modulation and channel polarization. To this end, we characterize the a priori distribution density as the sufficient statistic for the bit channel of individual modulation levels formed by an MLC scheme, in which the codeword of symbols is split into parallel bits streams. According to the factor graph generated by the binary PC, we apply the density evolution (DE) for each binary modulation level. Consequently, distribution densities of every bit within the modulated codeword can be obtained and we select the most reliable bits as information bits and set the complement as frozen bits. Comprehensive simulations demonstrate the validness of this approach compared to the benchmark scheme.

Polar codes are the first proven capacity-achieving codes for any symmetric binary input discrete memoryless channel (B-DMC) and their has been recent interest in their combination with the popular area of physical layer security. Previous work on error-correcting codes and physical layer security has assumed a wiretap channel where each user experiences Gaussian noise, but this paper presents the first results evaluating the performance of polar codes on wiretap channels where both the intended recipient and eavesdropper experience varying levels of impulsive noise. The impulsive noise has a symmetric alpha-stable probability density function (pdf) and density evolution is employed with knowledge of this pdf to construct optimal polar codes for an impulsive environment. Simulation results confirm that the bit error rate (BER) at the eavesdropper is always 0.5 for all signal-to-noise ratios, thus always ensuring secure communication, and we present the very interesting result where this is still achieved when the main channel is more impulsive than the eavesdropper's channel.

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.

For a number of quantum channels of interest, phase-flip errors occur far more frequently than bit-flip errors. When transmitting across these asymmetric channels, the decoding error rate can be reduced by tailoring the code used to the channel. However, analyzing the performance of stabilizer codes on these channels is made difficult by the #P-completeness of optimal decoding. To address this, at least for short codes, we demonstrate that the decoding error rate can be approximated by considering only a fraction of the possible errors caused by the channel. Using this approximate error rate calculation, we extend a recent result to show that there are a number of $[[5\leq n\leq12,1\leq k\leq3]]$ cyclic stabilizer codes that perform well on two different asymmetric channels. We also demonstrate that an indication of a stabilizer code's error rate is given by considering the error rate of a classical binary code related to the stabilizer. This classical error rate is far less complex to calculate, and we use it as the basis for a hill climbing algorithm, which we show to be effective at optimizing codes for asymmetric channels. Furthermore, we demonstrate that simple modifications can be made to our hill climbing algorithm to search for codes with desired structure requirements.

The neural successive cancellation (NSC) decoder with tanh-based modified log-likelihood ratio (LLR) is proposed for reducing the decoding latency of polar codes over free space optical (FSO) turbulence channel. The conventional successive cancellation (SC) decoder is partitioned into multiple sub-blocks, which are replaced by multiple sub neural network (NN) decoders with tanh-based modified LLR. The recursive characteristic of the polar sequences reliability ranking given in 5G standard enables the sub-NN decoder to be uniquely determined by code length and the number of information bits. Confirmed by the simulation, the bit error rate (BER) performance of NSC decoder with tanh-based modified LLR is close to the conventional SC decoder over turbulence channel for the practical-length polar codes. Regarding turbulence-stability, the NSC decoder trained in moderate and strong turbulence conditions have stable performance in a wide range of turbulence conditions. Moreover, in comparison of decoding latency, the NSC decoder with tanh-based modified LLR takes less than 25% time steps of SC decoder in the same code length.

The emerging visible light communication (VLC) technology, which has the advantages of short-distance communication and indoor lighting, has become a research hotspot of indoor wireless communication technology. Polar codes, as the only forward error correction (FEC) coding technology that has been theoretically proven to achieve Shannon's capabilities, are adopted to reduce the bit error rate (BER) in VLC systems. However, the individual polar codes are still unable to correct burst errors in VLC channels. This paper proposes concatenated polar codes with interleaving technology for VLC systems. The results show that it can effectively correct random errors and burst errors in VLC systems. Compared to the non-systematic polar codes, the systematic polar codes are more suitable as the inner code of the concatenated polar codes. The concatenated polar codes exhibit a lower error floor in VLC systems under burst errors than the individual polar codes. Compared with BCH-polar schemes, Convolutional-polar schemes have better robustness.

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.

Visible light communication (VLC) uses run-length limited (RLL) code to avoid flickering and support different dimming ranges. In this paper, we design an error control coding scheme for VLC using polar code as a forward error correction (FEC) code in serial concatenation with an RLL code. We propose to use soft iterative decoding between RLL code and polar code. Extrinsic information transfer (EXIT) chart is also used to illustrate the proposed serial concatenated scheme's iterative decoding behavior. We will show improvement in bit error rate (BER) performance as compared to the hard decoding based polar code proposed in the literature for VLC.

The increasing demand for high-density and high-speed digital optical recording systems has made the development of advanced coding and signal processing techniques for fast and reliable data recovery increasingly important. In recent years, the parity-check (PC)-code-based reception technique has been widely studied for magnetic recording systems, and it is projected to be highly promising for high-density optical recording. The PC code is an inner error correction code (ECC), which can detect dominant short error events of the system, using only a few parity bits. This reduces the loss in error correction capability of the outer ECC due to random short errors and results in a simple and efficient solution to improve the overall performance. This thesis is dedicated to the design and analysis of PCcode- based recording systems to achieve higher recording capacity with low implementation complexity, for high-density blue laser disc systems. In particular, most of the key components of the PC-code-based optical recording system have been designed and optimized for different recording densities, and different proportions of white noise and media noise. During the development of advanced coding and detection techniques, it becomes necessary to investigate the system’s performance with different coding schemes and recording densities. In Chapter 2 of the thesis, we propose a generalized Braat-Hopkins model for optical recording channels, which provides a fair basis for the performance comparison of detection schemes over different coding schemes and recording densities, for channels with additive, white Gaussian noise (AWGN) and media noise. Various basic issues associated with the PC-code-based systems are investigated in Chapter 3. These include bounds for bit error rates and error event probabilities, the dominant error events at channel detector output, different PC codes, simple and efficient post-processors, the impact of error events that are split across data block boundaries and the corresponding remedy, as well as the effects of code rates and recording densities. Simulation results show that a 4-bit PC code achieves the best performance. The corresponding bit error rates (BERs) are very close to the performance bounds, at both nominal and high densities. Constrained codes, which serve as a key component in the read-write channel of data storage systems, are desired to have a high code rate with low-complexity encoder/decoder. In Chapter 4, we investigate the design of certain capacity-approaching constrained codes for optical recording systems. In particular, we derive analytically the relationship between the number of encoder states and the maximum size of these codes. We identify the minimum number of encoder states that maximizes the code rate, for any desired codeword length. The design of constrained PC codes is key to the development of PCcode- based systems, and the systematic design of efficient constrained PC codes remains a challenging problem. In Chapter 5, we propose a general and systematic code design technique for constructing capacity-approaching constrained PC codes, which can detect any type of error events of the system, with the minimum code rate loss. Compared to the rate 2/3 code without parity, the newly designed constrained 4-bit PC code achieves a performance gain of 2 dB at nominal density, and 1.5 dB at high density, at BER = 10-5. As the dominant noise for high-density optical recording systems, media noise severely degrades the performance of channel detectors and postprocessors that are designed for AWGN. In Chapter 6, we propose two novel modifications to the bit detector to combat media noise. We further develop a data-dependent post-processing scheme for error correction. Compared to the system designed without considering media noise and without PC codes, the overall performance gain of the developed scheme can be more than 11 dB at high media noise levels. In data storage systems, the ECC failure rate (EFR) serves as the ultimate measure of the data recovery performance. In Chapter 7, we develop semi-analytical approaches for estimating the EFR, and analyze EFRs for the developed PC-code-based systems, for the cases without and with interleaving. Our analysis shows that with the optimum interleaving degrees, compared to the rate 9/13 code without parity, the 2-bit PC code achieves a gain of around 0.5 dB, and the 4-bit PC code gains 0.6 dB, at high record ing density and EFR = 10-16. The thesis concludes in Chapter 8 with some suggestions for further work.