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A long-standing vision of backscatter networks is to provide long-range connectivity and high-speed transmissions for batteryless Internet-of-Things (IoT). Recent years have seen major innovations in designing backscatter networks toward this goal. Yet, they either operate at a very short range, or experience extremely low throughput. This paper takes one step further towards breaking this stalemate, by presenting PolarScatter that exploits channel polarization in long-range backscatter networks. We transform backscatter channels into nearly noiseless virtual channels through channel polarization, and convey bits with extremely low error probability. Specifically, we propose a new polar code scheme that automatically adapts itself to different channel quality, and design a low-cost encoder to accommodate polar codes on resource-constrained backscatter tags. Furthermore, we devise a new metric to calculate log-likelihood ratio for accurate decoding, and present a stopping criterion of iterations to reduce decoding latency. We build a prototype PCB tag, and our experiments show that it achieves up to 11.5 throughput improvement over the state-of-the-art long-range backscatter solution. We also simulate an IC design in TSMC 65 nm LP CMOS process. Compared with traditional encoders, our encoder reduces storage overhead by three orders of magnitude, and lowers the power consumption to tens of microwatts.

A highly secure and reliable optical orthogonal frequency division multiplexing passive optical network (OFDM-PON) based on the chaos encryption inside polar code scheme is proposed. The 3-dimensional Chua's circuit model is adopted to realize chaotic encryption during polar encoding with low complexity. The three chaotic sequences generated by Chua's circuit model are used to encrypt information index bit matrix, frozen bits (during polar encoding) and subcarrier sequence, respectively. The BER and security performance of the optical transmission system are enhanced by our method. A 70 Gb/s polar encoded encrypted 16 quadrature amplitude modulation OFDM signal transmission over a 2 km seven-core fiber is further experimentally demonstrated. The experimental results show that the polar encoded encrypted 16 QAM-OFDM obtains a 2.1 dB gain in receiver sensitivity at a BER of 10−3 under the same bit rate compared to the normal 16 QAM-OFDM without FEC. Moreover, the key space of the proposed scheme is 1084, which is large enough to guard against any malicious attacks from illegal ONUs effectively. The combined superiority of BER and security performance enable such OFDM-PON based on chaos encryption inside polar code, to have a high prospect of application in low-cost and reliable optical access system.

Polar codes have been receiving increased attention for application in beyond 5G networks. They offer low-complexity decoding algorithm and can achieve symmetric channel capacity. However, the majority of research works have focused on the codes constructed by the binary kernel (2×2 polarization matrix) which bounds the code length to an integer power of 2. Multi-kernel polar codes have been proposed as a method that allows the construction of polar codes with sizes different from powers of 2 by mixing multiple kernels of different dimensions. A hardware implementation based on the successive cancellation (SC) algorithm found in the literature shows that it suffers from a long decoding latency. In this paper, we design and implement a multi-kernel decoder based on the fast-simplified SC (fast-SSC) algorithm to decrease the decoding latency. It can decode any code constructed by binary and ternary (3×3) kernels featuring flexible code length, code rate, and kernel sequence. FPGA implementation results reveal that a polar code of length N = 1536, rate R = 1/2 with Processing Element (Pₑ) value of Pₑ = 240, gains 84.6% lower latency compared to the original algorithm. Also, the architecture supports polar codes constructed by purely-binary and purely-ternary kernels. A polar code of length N = 1024, rate R = 1/2, and Pₑ =120 achieves an information throughput of 432 Mbps

For an orthogonal frequency division multiplexing (OFDM) communication system which requires channel estimation for each sub-carrier without using pilot signals, there may exist the problem of phase ambiguity. By extending the conventional concept of noncoherent block coding (NBC), we propose a $Q$ -section polar code that has the NBC property in such a way that each codeword is divided into $Q$ sections and reversing the code bits in some sections results in another codeword which is encoded by the same desired message. The polar coded OFDM with a $Q$ -section NBC property enables us to solve the problem of phase ambiguity that exists in each sub-carrier. In addition, the $Q$ -section NBC property also enables us to employ the multiple-candidate method for reducing the peak-to-average-power ratio (PAPR).

How to efficiently rank the reliability order of the channels is critical for polar code construction. A new polar code construction framework tailored to a given decoding algorithm based on the genetic algorithm has been proposed to achieve an error correction performance gain. However, the time complexity of this scheme is extremely high. In this letter, a scheme based on a hash table is proposed to generally reduce the computational complexity of the repeated candidates in the genetic algorithm-based system. A two-step method is proposed to improve the crossover process when the system has prior knowledge. It is shown from the numeric results that the proposed scheme achieves computational complexity reduction of the repeated candidates. An error correction performance gain is achieved over the original scheme under successive cancellation list decoding.

In this letter, a double polar code (D-Polar) for joint source and channel coding (JSCC) is proposed, in which the source compress is implemented by a polar code, the channel error correction is performed by a systematic polar code. Furthermore, a turbo-like belief propagation (TL-BP) decoder consisted of a channel BP decoder and a source BP decoder is proposed for joint source and channel decoding in the receiver. In this TL-BP decoder, the soft information is exchanged between the channel BP decoder and the source BP decoder so as to improve the efficiency of channel decoding in utilizing the source information residual in the compressed bits. Simulation results show that the performance of the proposed D-Polar JSCC scheme with TL-BP decoder is significantly improved compared with the existing source-channel joint polarization scheme.

For polar codes, the statistical breakdown of belief propagation (BP) decoding errors is firstly proposed by Sun et al. The errors are classified into three categories: unconverged errors, false converged errors and oscillation errors, which are corrected by three different post-processing methods. As for the BP List (BPL) decoding, we discover that one type of error can be transformed to another type by different factor graphs in the failed BP decoding. Besides, false converged error is usually easier to be detected and modified. Therefore, we only target false converged error to detect and correct. In this work, we propose a two-level detection rule for false converged errors and its corresponding post-processing algorithm to modify these errors in order to improve the performance of the BPL decoding. Numerical results show that the error-correction performance of the proposed decoder is more than 0.85dB better than that of the permuted BPL (PBPL) decoder with slight extra computation complexity at the frame error rate (FER) of $10^{-5}$ for 5G (1024,512) polar code.

The increasing data traffic rate of wireless communication systems forces the development of new technologies mandatory. Providing high data rate, extremely low latency and improvement on quality of service are the main subjects of next generation 5G wireless communication systems which will be in the people's life in the years of 2020. As the newest and first mathematically proven forward error correction code, polar code is one of the best candidates among error correction methods that can be employed for 5G wireless networks. The aim of this tutorial is to show that belief propagation decoding of polar codes can be a promising forward error correction technique in upcoming 5G frameworks. First, we survey the novel approaches to the belief propagation based decoding of polar codes and continue with the studies about the simplification of these decoders. Moreover, early detection and termination methods and concept of scheduling are going to be presented throughout themanuscript. Finally, polar construction algorithms, error types in belief propagation based decoders and hardware implementations are going to be mentioned. Overall, this tutorial proves that the BP based decoding of polar codes has a great potential to be a part of communication standards.

In the present article, we propose a novel selected mapping (SLM) scheme based on a Polar coding technique for peak to average power ratio (PAPR) reduction in orthogonal frequency division multiplexing with index modulation (OFDM-IM) systems. The proposed scheme by utilizing random frozen bits in polar codes generates different number of candidate sequences. Then the candidate sequence with the smallest PAPR is selected for transmission. The proposed scheme can be considered as one block in an OFDM-IM system that performs both PAPR reduction and error correction. For a fair comparison, the performance of an OFDM-IM system based on the proposed scheme is compared with a system that carries out the same operations with two separate blocks where a Polar encoder is cascaded with the one for PAPR reduction (such as conventional SLM or PTS), and a Polar coded OFDM-IM system without any PAPR reduction scheme. Moreover, based on our proposed SLM scheme, a novel receiver without using side information (SI) is proposed. This SI free receiver is based on the successive cancellation list (SCL) polar decoding scheme. Simulation results show that as the Polar code can bring both error correction and PAPR reduction capabilities by using our proposed scheme. Also, the proposed SI free receiver can achieve similar bit error rate (BER) performance as that of the ideal case in both additive Gaussian white noise (AWGN) and frequency selective channels.

Achieving the attractive error-correcting capability with a simple decoder structure, the polar code using successive cancellation (SC) decoding is now expected to be installed at the resource-limited IoT or embedded communications. However, the existing SC decoders normally suffer from the long processing latency caused by the serialized processing steps, limiting the practical applications of polar codes. In this article, to solve this latency problem, we present a new low-complexity merging operation that can increase the number of parallel factors for realizing the tree-level parallelism. We also modify the previous pruning method to further reduce the number of visited nodes at the parallel SC decoding scenario. In addition, a novel parallel partial-sum calculator (PSC) architecture is introduced to update partial-sum registers with multiple decoded bits by taking only one processing cycle. Implementation results show that the proposed 8-parallel SC polar decoder in 28-nm CMOS requires only 0.140 $\mu \text{s}$ to decode a (1024, 512) codeword of 5G system, remarkably reducing the decoding latency when compared to the state-of-the-art designs.

Polar codes are capacity-achieving channel codes and they have recently been adopted for fifth-generation (5G) enhanced mobile broadband (eMBB) control channels. Using successive cancellation list (SCL) decoding, the error-correction performance of polar codes can surpass state-of-the-art codes of a comparable length. However, the sequential SC decoding incurs a long latency, and list decoding requires complex tracking of candidates. We present a split-tree SCL decoder that works by dividing a polar code’s decoding tree to sub-trees following a split-tree decoding algorithm. The sub-trees are decoded in parallel by smaller sub-decoders that reconcile their decisions in every decoding stage. The split-tree list decoder architecture improves the throughput and latency proportionally to the split factor. By exploiting under-utilized hardware resources, we apply frame interleaving to further increase throughput and employ dynamic clock gating to reduce energy. The results are demonstrated in a 0.64-mm2 40-nm test chip that implements a split-4, list-2, eight-frame-interleaved decoding architecture. The chip supports configurable code lengths up to 1024 bit and variable code rates. At 0.9 V and room temperature, the chip achieves 3.25 Gb/s with 42.8-mW power, or 13.2 pJ/b, and demonstrates competitive error-correction performance.

Polar code is a kind of capacity-approaching code with explicit structure as part of the next generation wireless communication standard. The performance of the polar code is directly determined by the construction method, in which there are three important parameters: code rate $(R)$ , block-length $(N)$ , and designSNR ( DSNR ). It is very difficult to obtain an optimal construction because all these three parameters should be optimized in a recursive structure model. In this article, an efficient method, which constructs the desired optimal polar code by simultaneously selecting the three parameters in terms of the error probability, is presented. In the proposed construction, a dimensionality reduction search method is used to solve a fore mentioned optimization problem and to find the relation between the performance of polar code and design parameters. At the same time, the optimal parameters $N$ and $R$ can be obtained under any DSNR by taking the average group accuracy rate as the evaluation criterion. Through this method, the appropriate solution of three parameters in the construction of the polar code can be determined according to the required error probability. The proposed construction method is verified by numerical simulation, and the optimal performance to $N$ , $R $ and DSNR is compared and discussed.

In this paper, we apply polar codes to a modified version of OFDM systems with index modulation which is called OFDM with in-phase/quadrature index modulation (OFDM-I/Q-IM). We provide general design guidelines for the proposed polar-coded OFDM-I/Q-IM systems. In the proposed system, at the transmitter, we employ a random frozen bits appending scheme which not only makes the polar code compatible with OFDM-I/Q-IM but also improves the bit error rate (BER) performance of the system. Furthermore, at the receiver, it is shown that the a posteriori information for each index provided by the index detector is essential for the iterative decoding of polar codes by the belief propagation (BP) algorithm. Simulation results show that the proposed polar-coded OFDM-I/Q-IM system outperforms its OFDM counterpart in terms of BER performance.

In this paper, we introduce a structural extension method of polar codes as a new code modification method. Structural extension is a systematic method of generating additional codeword bits from the mother code by selecting intermediate bits obtained in the middle of the polar encoding. We start by studying simplex-extension from a $2\times 2$ local polarization kernel, and then, we generalize this method to construct polar codes of arbitrary lengths. This extension method does not change the original polar code structure, and thus, the conventional encoder and decoder can be employed with a slight modification. We present a code construction tool by modifying density evolution to design punctured, shortened, and extended polar codes. By analysis and numerical experiments, we show that the proposed structural extension method enables to achieve an excellent tradeoff between performance and implementation complexity.

A method for construction of polar subcodes with reduced error coefficient is presented. The proposed approach relies on explicit enumeration of low-weight non-zero codewords in a polar code, and construction of dynamic freezing constraints which define a subcode not containing most of these codewords. The obtained codes provide a large performance gain in the high-SNR region compared to non-optimized polar subcodes and polar codes with CRC.

Traditional polar codes are constructed based on a size-2 kernel and the code length is thus restricted to an integer power of 2. To achieve flexible code lengths, punctured and shortened schemes are used in the existing 5G standard. Recently, an alternative approach for flexible code lengths, multi-kernel (MK) polar code, has been proposed, which is constructed based on both size-2 and other larges kernels. However, the error-correction performance is not as good as that of the punctured and shortened schemes. In this letter, an MK polar code construction method based on a kernel substitution scheme is proposed, which is a practical scheme to construct the codes using two types of large kernels with the same size so that the error-correction performance can be improved. Compared with the existing MK polar codes, simulation results show that 0.15 - 0.25dB performance gain can be achieved by SC decoding for most code settings at a block error rate of 10−4 without complexity overhead.

In this paper, we investigate physical (PHY) layer message authentication to combat adversaries with infinite computational capacity. Specifically, a PHY-layer authentication framework over a wiretap channel ( $W_1,W_2$ W 1 , W 2 ) is proposed to achieve information-theoretic security with the same key. We develop a theorem to reveal the requirements/conditions for the authentication framework to be information-theoretic secure for authenticating a polynomial number of messages in terms of $n$ n . Based on this theorem, we design an authentication protocol that can guarantee the security requirements, and prove its authentication rate can approach infinity when $n$ n goes to infinity. Furthermore, we design and implement a feasible and efficient message authentication protocol over binary symmetric wiretap channel (BSWC) by using Linear Feedback Shifting Register based (LFSR-based) hash functions and strong secure polar code. Through extensive simulations, it is demonstrated that the proposed protocol can achieve high authentication rate, with low time cost and authentication error rate.

High peak-to-average power ratio (PAPR) has been major drawback of orthogonal frequency-division multiplexing (OFDM). To reduce the PAPR in the polar-coded OFDM systems, we propose a polar coding-based selective mapping (SLM) which does not transmit redundant information. Based on a set of independent frozen data blocks, we construct coset codes of a linear polar code to efficiently generate multiple OFDM signals, and selectively transmit the one with the lowest PAPR. Since the proposed SLM does not transmit either the side information or PAPR control labels, we can avoid the data rate loss. We also present an adaptive trial-based blind receiver to perform not only decoding of the unfrozen data block but also detection of the selected signal (or frozen data block). Simulation results show that the PAPR performance of the proposed SLM can approach to the optimal, and the blind receiver can support the proposed SLM with high accuracy.

In the paper, bit-interleaved polar-coded modulation discrete multi-tone (DMT) (BIPCM-DMT) with 256-QAM enabled by constant amplitude zero auto-correlation (CAZAC) precoding is proposed and experimentally demonstrated in a low-cost intensity-modulation and direct-detection (IM-DD) system. In order to conquer the fading effect on mapped bit-levels, two types of interleaver including quadratic polynomial permutation (QPP) interleaver and random interleaver are compared to optimize the system performance. However, for 256-QAM BIPCM-DMT system, the signal-to-noise ratio (SNR) fades dramatically as the data subcarrier index increases, it is difficult to match the well-designed operating point of Monte Carlo in the process of polar code construction. Thus, the precoding schemes including CAZAC and orthogonal circulant transform (OCT) are proposed to equalize the uneven SNRs in BIPCM-DMT system. Compared with the bit / power loading and pre-equalization schemes, it will greatly reduce the cost and complexity of the system because channel information obtained by the transmitter via round-trip feedback is not considered. In addition, CAZAC precoding is selected to optimize the BIPCM-DMT system because it is superior to OCT precoding in reducing PAPR. The experimental results show that aided by CAZAC precoding, 20.63 Gb/s BIPCM-DMT signal can be received with an error-free over 50-km standard single-mode fiber (SSMF). Furthermore, compared with BIPCM-free, only CAZAC precoding, and only random interleaver schemes, the proposed scheme can achieve Q-factor enhancement of 4.82 dB, 2.46 and 1.78 dB, respectively, at the received optical power (ROP) of −5 dBm.

Military robot swarm becomes more and more important for contemporary military application. In order to satisfy sophisticated application scenarios, novel communication architecture for military robot swarm has to be conceived. In this paper, we propose a multi-carrier intelligent covert satellite communication (MCICSC) architecture based on electromagnetic environment sensing technology, in where the ability of dynamically adapting the system configuration to the holistic requirements of covertness, performance, and complexity can be considerably improved. Additionally, in order to solve the carrier synchronization problem in very low signal noise ratio (SNR) for each data stream of MCICSC, polar code (PC) is incorporated into the proposed MCICSC system. With the aid of the proposed polar code-aided multi-stream parallel feedback (PAMSPF) carrier synchronization scheme, the MCICSC system is capable of realizing high-precise carrier synchronization even in very low SNR region. Moreover, in order to further decrease the complexity of our MCICSC system, we also propose a polar code-based diversity combining (PBDC) scheme, which is capable of efficiently combing all the data streams without the aid of channel state information (CSI) estimation. Numerical simulations demonstrate that for MCICSC system the performance of the PAMSPF carrier synchronization and PBDC combining schemes can get extremely close to that of an ideally synchronized system. Complexity analysis of the proposed algorithms is given as well.

The successive-cancellation flip (SCFlip) decoder and its variants provide a significant coding gain with the average complexity practically identical to that of the successive cancellation (SC) decoder in a wide range of signal-to-noise ratios (SNRs). But, they suffer from high complexity and long latency when the SNR decreases, since the average number of extra decoding attempts becomes inevitably large. To mitigate this problem, we propose a novel SCFlip decoder, called an error-aware SCFlip (EA-SCFlip) decoder, for distributed cyclic-redundancy-check (CRC) polar codes. Based on the distributed CRC bits, it employs early termination at each extra decoding attempt so that it can reduce the decoding complexity and latency on the average. It also reduces the search space of candidate bit-flips in the dynamic building of the bit-flip list by exploiting the parity-check relationship (PCR) of the first error-detected CRC bit at each extra decoding attempt. Furthermore, we propose a greedy algorithm to design a distributed CRC code such that the obtained PCRs make the early-error-detection capability of the EA-SCFlip decoder as high as possible. Numerical results demonstrate that the EA-SCFlip decoder can indeed achieve an early termination gain as well as a complexity reduction, when a polar code is concatenated with the distributed CRC code designed by the proposed algorithm.

In this article we analyze the reliability of the synthetic channels of a polar code designed for the binary erasure channel, and in particular the properties of their Bhattacharyya parameter. We apply techniques from network theory in order to establish fresh results. Firstly, we show how to efficiently approximate the Bhattacharyya parameter. Secondly, we identify asymptotically “good” channels for polar code. Thirdly, we determine a range of values for the erasure probability for which the selecting rule for the polar codes construction coincides with that of the Reed-Muller codes.

As the fifth-generation (5G) wireless networks’ key technology, the joint design of SCMA and polar code is concerned by the future communication system. In this paper, a CRC aided joint iterative detection and successive cancellation list (SCL) decoding (CAJIDS) receiver is proposed for the polar coded SCMA (PC-SCMA) system. For the receiver, the SCL decoder’s extrinsic messages construction algorithm is designed by using Bayes rule and soft cancellation algorithm. Additionally, the distributed CRC aided polar (DCA-polar) code and variable list size are used to reduce the receiver’s decoding latency and complexity. Simulation results demonstrate that the CAJIDS receiver has better error rate performance than the joint iterative detection and decoding (JIDD) receiver. It also outperforms the LDPC coded SCMA (LDPC-SCMA) system. Specifically, when code length $N=256$ and code rate $R = 1/3$ , the CAJIDS outperforms the JIDD and LDPC-SCMA 1.2 dB and 1.9 dB over the Rayleigh fading channel, respectively. It also shows that, compared with the fixed-list-size receiver, with a similar error rate performance, the CAJIDS receiver has lower decoding latency and complexity.

Interest in polar codes has increased significantly upon their selection as a coding scheme for the 5th generation wireless communication standard (5G). While the research on polar code decoders mostly targets improved throughput, few implementations address energy consumption, which is critical for platforms that prioritize energy efficiency, such as massive machine-type communications (mMTC). In this work, we first propose a novel Fast-SSC decoder architecture that has novel architectural optimizations to reduce area, power, and energy consumption. Then, we extend our work to an energy-efficient implementation of the fast SC-Flip (SCF) decoder. We show that sorting a limited number of indices for extra decoding attempts is sufficient to practically match the performance of SCF, which enables employing a low-complexity sorter architecture. To our knowledge, the proposed SCF architecture is the first hardware realization of fast SCF decoding. Synthesis results targeting TSMC 65nm CMOS technology show that the proposed Fast-SSC decoder architecture is 18% more energy-efficient, has 14% less area and 30% less power consumption compared to state-of-the-art decoders in the literature. Compared to the state-of-the-art available SC-List (SCL) decoders that have equivalent error-correction performance, proposed Fast-SCF decoder is 29% faster while being $2.7\times $ more energy-efficient and 51% more area-efficient.

The belief propagation (BP) decoding of polar codes provides better bit error rate (BER) and packet error rate (PER) performance than the successive cancelation (SC) decoding. It has been reported in the literature that the graph of BP decoding of polar codes can be permuted to provide even better decoding performance. In this paper, we theoretically prove that all permutations of the graph are equivalent in terms of the polar encoding. For BP decoding, it is shown that permutations of the layers of the graph also permute the bit channels. A principle which protects the weakest information bit channel is provided in the paper. Two parallel BP decoders, one employing the standard graph and the other selected based on the selection principle, are also proposed to work together. The simulation results show that when the total number of iterations is fixed, the proposed BP decoding can have the same performance as the existing permutation scheme while provides a higher throughput.

We experimentally demonstrate the transmission of orthogonal frequency division multiplexing (OFDM) signals with polar codes FEC in a W-band MMW system. The BER performances of the M-QAM OFDM signal with FEC and without FEC are investigated, respectively. The experimental results show that the polar codes have a net gain of 7.9 dB compare with that of the LDPC code (7.4 dB). To the best of our knowledge, this is the first demonstration that the polar codes FEC are used in a W-band MMW system with OFDM modulation.

Polar codes have been adopted as the coding scheme for control channels in the 5G communication system, where the blind detection is required. In this letter, to mitigate the false-alarm rate (FAR) in the polar code blind detection, we propose a more efficient architecture which combines the conventional list decoding and a binary classifier. After the list decoding, the CRC-passing result will be further checked by the classifier to determine whether the received signal is the intended control information or not. The classifier works by inspecting the squared Euclidean distance ratio (SEDR) between the received signal and the hypotheses, and the decision threshold used in the classifier is automatically learnt by a neural network (NN). The presented results show that very short CRC sequences can be enough to reach the target FAR, and this CRC overhead reduction finally contributes to the performance improvement of polar codes.

A novel joint channel coding and peak-to-average power ratio (PAPR) reduction technique for polar-coded orthogonal frequency-division multiplexing (OFDM) systems is proposed. The main concept is to introduce shaping indices to polar encoder, and generate multiple OFDM symbol candidates according to various shaping bit patterns and then select the one with the lowest PAPR as a transmitted signal. Since information and shaping bits can be jointly decoded using the conventional decoding algorithm of polar codes, transmission of side information is not required. The polar code design algorithm for efficient PAPR reduction is also discussed. It is demonstrated that the proposed shaping method offers a gain of more than 4.5dB in terms of PAPR performance without affecting the decoding complexity.

As an improved version of successive cancellation (SC) polar decoder, an SC stack (SCS) decoder has been proposed for performance improvement. However, the existing SCS polar decoder suffers a lot from high time complexity at low signal-to-noise ratio (SNR) region and space complexity compared with the SC decoder. To this end, two improved decoders are proposed to reduce time and space complexity in both low and high SNR regions. The first one is the segmented cyclic redundancy check (CRC)-aided SCS (SCA-SCS) decoder, which is based on segmented parity checkers. The second one is the adaptive SCS (ASCS) decoder, which has the flexibility of stack depth and searching width. Furthermore, a channel condition estimator is proposed to select appropriate decision criteria for different SNR scenarios. Results have shown that for the polar code of length 1024 and rate 1/2, two improved SCS decoders can perform better than the traditional SCS decoder. The proposed SCA-SCS decoder and the ASCS decoder can achieve 10.8% and 11.42% time complexity reduction and 31.68% and 60.85% space complexity reduction on average over binary-input additive white Gaussian noise channels (BI-AWGNCs), respectively. Efficient parallel hardware architecture of the SCS polar decoder is first proposed and implemented with 90- and 65-nm technologies. Results have verified its advantages over the state of the art (SOA).

The probabilistic shaping scheme by Honda and Yamamoto (2013) for polar codes is used to enable power-efficient signaling for on–off keying (OOK). As OOK has a non-symmetric optimal input distribution, shaping approaches that are based on the concatenation of a distribution matcher followed by systematic encoding do not result in optimal signaling. Instead, these approaches represent a time-sharing scheme, where only a fraction of the codeword symbols is shaped. The proposed scheme uses a polar code for joint distribution matching and forward error correction which enables asymptotically optimal signaling. Numerical simulations show a gain of 1.8 dB compared to uniform transmission at a spectral efficiency of 0.25 bits/channel use for a blocklength of 65 536 bits.

We study partial orders (POs) for the synthesized bit-channels of polar codes. First, we give an alternative proof of an existing PO for bit-channels with the same Hamming weight and use the underlying idea to extend the bit-channel ordering to some additional cases. In particular, the bit-channel ordering for a given code block length is used to generate additional bit-channel ordering relationships for larger block lengths, generalizing previously known POs. Next, we consider POs especially for the binary erasure channel (BEC). We identify a symmetry property of the Bhattacharyya parameters of complementary bit-channel pairs on the BEC and provide a condition for the alignment of polarized sets of bit-channels for the BEC and general binary-input memoryless symmetric (BMS) channels. Numerical examples and further properties about the POs for the bit-channels with different Hamming weights are provided to illustrate the new POs. The bit-channels with universal ordering positions, which are independent of the channel erasure probability, are verified for all of the code block lengths. Finally, we show the threshold behavior of the Bhattacharyya parameters of some bit-channels by approximating the threshold values. The corresponding value for a bit-channel can be used to determine whether it is good or bad when the underlying channel is known.

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.

The successive-cancellation list (SCL) and successive-cancellation flip (SCF) decoding can be used to improve the performance of polar codes, especially for short to moderate length codes. However, their serial decoding nature results in significant decoding latencies. Implementing some operations in parallel can reduce their decoding latencies. This paper presents fast implementations of the SCL and SCF decoders. In particular, we propose fast parallel list decoders for five newly identified types of nodes in the decoding tree of a polar code, which significantly improves the decoding latency. We also present novel fast SCF decoders that decode some special nodes in the decoding tree of a polar code without serially computing bit log-likelihood ratios. Using our proposed fast parallel SCF decoders, we observed an improvement up to 81% with respect to the original SCF decoder. This significant reduction in the decoding latency is observed without sacrificing the bit-error-rate performance of the code.

For a coded massive multi-user multiple-input multiple-output (MIMO) system, a soft-output MIMO detector is essential since it can provide a significant coding gain, e.g., 3 dB, compared with a hard-output detector. However, the computational complexity of the soft-output MIMO detector is usually much greater than that of the hard-output detector. This paper presents the first soft-output message-passing detector (MPD), which is also integrated with a high-throughput polar decoder. The algorithm and architecture are designed concurrently to improve the hardware performance. The proposed techniques, including the adaptive variance estimation and reliable symbol detection, reduce the complexity of the MPD by 98.3%, and enable soft output for the outer polar decoder. Compared to the state-of-the-art hard-output design, the proposed MPD achieves a $5.86\times $ higher throughput-to-area ratio (TAR) with 54.3% lower energy dissipation, despite the soft output. The proposed bidirectional-propagation belief propagation decoder is devised to reduce the critical path and to increase the throughput. The proposed polar decoder can improve the TAR by 35% with a comparable area and energy compared with the state-of-the-art polar decoder. A polar-coded massive MIMO receiver that supports a length-1024 rate-1/2 polar code, 128 receive antennas, and eight users is designed and implemented, and delivers a throughput of 7.61 Gb/s.

Internet of Vehicles (IoV) is the basis of future intelligent transportation systems. Both the control signaling and data dissemination services in IoV must be transmitted with high reliability and low latency so that safety can be guaranteed. Based on the discussion and analyses of issues in achieving high reliability low latency transmissions, we present in this paper a Polar code-based overlapping transmission scheme in which simultaneous transmissions from different nodes to the same receiving node are allowed to use the same time-frequency resource block. To effectively eliminate multiple access interference introduced by the overlapped nonorthogonal transmissions, a successive cancellation list-based improved interference elimination decoding algorithm (SCL-based IIEDA) is proposed to retrieve the Polar coded information. Numerical results show that the SCL-based IIEDA performs well on information recovery in the presented transmission scheme. In addition, it is shown that the proposed scheme not only significantly reduces the acknowledgment overhead but also greatly improves the spectral efficiency.

In this paper a brand new channel error detection and correction method is provided. Artificially adding symbols to the source binary sequence makes the binary sequence present a regularity. The receiver can use this rule to implement error detection and correction. Since many redundant symbols are added, a weighted probability model lossless coding method is proposed in this paper to remove redundant information, and the reasons why Markov chains and conditional probabilities are not feasible are given. It is proven that the method in this paper can reach the channel capacity when the code length approaches infinity. It is shown experimentally that when the code rate is 1/2 in the BIAWGN channel, the method in this paper is superior to the polar code and LDPC code.

A polar code with repeating bits (RB polar code), where a part of frozen bits only repeat their preceding information bits, is proposed. In the code construction, we propose cluster pairwise error probability (CPEP) to measure the competitiveness of the correct path against the error paths and apply CPEP as the optimization criterion for the code construction, aiming to reduce the elimination probability of the correct path in a successive cancellation list (SCL) decoder. In the simulations, the constructed RB polar codes are combined with the cyclic redundancy check (CRC) code and outperform their counterparts of CRC-concatenated polar codes over various codeword lengths, code rates, and puncturing patterns.

Polar codes were first proposed by E. Arıkan in 2009 and have received significant attention in recent years. Successive-cancellation (SC) and belief-propagation (BP) decoding algorithms have been applied by some researchers to polar codes. However, unlike SC-based decoders, the performance optimization of BP-based decoders has not been fully explored yet, especially in regard to the impact of the number of iterations on the decoding complexity. In this paper, a novel early stopping criterion based on partial frozen bits for belief-propagation polar code decoders is designed. The proposed criterion is based on the fact that some of the frozen bits that are known to the decoders have a higher average error probability than the information bits and can be used to terminate the decoding. Furthermore, the hardware architecture of the BP-based polar code decoder with the proposed stopping criterion is presented. The simulation results show that the proposed early stopping criterion greatly reduces the number of iterations of BP-based polar code decoders without any performance loss and reduces the hardware complexity from O (NlogN) to O (N) compared with state-of-the-art design.

A multi-cyclic redundancy check (Multi-CRC) polar code construction algorithm is proposed in this paper to solve the error propagation problem of successive cancellation decoding for polar codes. In this algorithm, the information sequence is optimized into several segments to allow decoding errors to be corrected in time, minimizing the impact of error propagation. An improved multi-successive cancellation bit flipping (M-SCFlip) decoding algorithm is proposed to execute the bit flipping operation after CRC check-in each segment. In the low-SNR region, the proposed new multi-CRC polar code with successive cancellation list (SCL) decoding has a slight frame-error rate (FER) degradation compared with the original CRC polar code. With the M-SCFlip decoding algorithm developed in this paper, it achieves a better FER performance compared with the CRC polar code with successive cancellation (SC) and SCL ( $L\!=\!2$ ) decoding algorithms. In addition, it has a lower decoding delay and requires a lower memory space. For example, at a FER of 10−4 with the same code length and effective code rate, the proposed multi-CRC polar code with M-CFlip decoding achieves a 1.19 dB and 0.79 dB gains over existing CRC polar codes with the SC and SCL ( $L\!=\!2$ ) decoding algorithms, respectively.

We propose a two-phase Monte Carlo (TPMC) method to accelerate the original Monte Carlo (MC) method for constructing polar codes with high-dimensional kernels. In the TPMC method, some of the most reliable and unreliable bits are obtained by Gaussian approximate-density evolution (GA-DE) method in the first phase; in the second phase, these most reliable and unreliable bits are viewed as frozen bits. Then, the MC method are used to evaluate the remaining bits and select some best bits from the remaining bits. Finally, these best bits and the most reliable bits are combined as the information bits of the constructed polar code. By our investigation, most bits can be fixed as frozen bits in the second phase without error performance loss of the constructed polar codes. Because computation of frozen bits can be saved and computation of the GA-DE method can be ignored in contrast to that of the MC method, the TPMC method substantially reduces the complexity of the MC method. Simulation results show that 1) For a G⊗315 polar code with block length 3375 and code rate 1/2, the TPMC method can fix 3200 bits as frozen bits in the second phase without error performance loss, which reduces the computation cost by approximately 92.6% over the MC method; 2) Polar codes with high-dimensional kernels constructed by the TPMC method outperforms polar codes with the G2 kernel constructed by the Tal-Vardy method in terms of error performance; 3) With the same computational cost, the TPMC method can construct better polar codes than the MC method.

We study an uplink multiuser multiple-input multiple-output (MU-MIMO) system with one-bit analog-to-digital converters (ADCs) in which one base station (BS) with Nr receive antennas serve K users with a single antenna. For this system, the soft-output (SO) detector was recently proposed where a softmetric (e.g., a log-likelihood ratio (LLR)) is computed from a hard-decision channel output by introducing a novel distance measure in the binary Hamming space. This makes it possible to be naturally incorporated into the state-of-the-art channel codes (e.g., low-density parity-check code or polar code). In this paper, we further improve the performance of the SO detector by exploiting a priori information (e.g., the previously decoded messages), which is called the one-bit successive-cancellation soft-output (OSS) detector. The key idea of the proposed OSS detector is that each user k's message is decoded sequentially via the associated channel decoder k in ascending order and a refined search-space is constructed using the previously decoded messages (i.e., the enhanced LLRs are generated). We then present a multiple OSS detector by taking into account a more practical scenario where the BS is equipped with multiple channel decoders. In addition, we propose an efficient way to determine a good decoding order by introducing a novel set-distance measure. Finally, simulation results demonstrate that the proposed OSS detector can significantly improve the existing SO detector for the coded MU-MIMO systems with one-bit ADCs.

Polar coded sparse code multiple access (SCMA) system is conceived in this paper. A simple but new iterative multiuser detection framework is proposed, which consists of a message passing algorithm (MPA) based multiuser detector and a soft-input soft-output (SISO) successive cancellation (SC) polar decoder. In particular, the SISO polar decoding process is realized by a specifically designed soft re-encoder, which is concatenated to the original SC decoder. This soft re-encoder is capable of reconstructing the soft information of the entire polar codeword based on previously detected log-likelihood ratios (LLRs) of information bits. Benefiting from the soft re-encoding algorithm, the resultant iterative detection strategy is able to obtain a salient coding gain. Our simulation results demonstrate that significant improvement in error performance is achieved by the proposed polar-coded SCMA in additive white Gaussian noise (AWGN) channels, where the performance of the conventional SISO belief propagation (BP) polar decoder aided SCMA, the turbo coded SCMA and the low-density parity-check (LDPC) coded SCMA are employed as benchmarks.

In this letter, we construct rate compatible polar codes via polarizing matrix extension, to support efficient adaptive incremental redundancy hybrid automatic repeat request (IR-HARQ) protocol. We show that an arbitrary number of incremental coded bits can be generated by extending the polarizing matrix, such that multiple retransmissions are aggregated to produce a longer polar code with extra coding gain. This coding gain is maximized by selecting the most reliable bit channels in the extended codes, rather than the shorter codes, as information bits. Naturally, some information bits are transferred from the shorter part to the extended part for higher bit channel reliability. One-to-one parity-checking between the two parts lends itself well to the above purpose during the construction of the longer code. Simulation results demonstrate consistent coding gain over the chase combining HARQ scheme.

Bit-channels of finite-length polar codes are not fully polarized, and the negative influences from semi-polarized channels are not negligible. In this letter, we consider the concatenated system of an inner polar code and an outer low-density parity-check (LDPC) code to improve the error correction performance of finite-length polar code with belief propagation (BP) decoding. We propose a bit mapping method between LDPC coded bits and polarized channels, aiming to match the coding protection of LDPC codes with polarization of polar codes. Simulation results show that, the concatenated polar codes with bit mapping outperform the pure polar codes with BP decoding and concatenated polar codes with traditional consecutive bit mapping.

This letter proposes a reinterpretation of the classical binary Golay codes as Polar codes with additional inner permutations and puncturing. This establishes a novel relationship between algebraic codes and codes based on Kronecker products, like polar codes. Naturally, this construction also allows for polar-code-type decoding of the Golay codes using successive cancellation list decoding. Using the Golay code as a driving example, we conjecture that other algebraic codes may be represented in a similar way.

SCMA and polar coding are possible candidates for 5G systems. In this paper, we firstly propose the joint iterative detection and decoding (JIDD) receiver for the uplink polar coded sparse code multiple access (PC-SCMA) system. Then, the EXIT chart is used to investigate the performance of the JIDD receiver. Additionally, we optimize the system design and polar code construction based on the EXIT chart analysis. The proposed receiver integrates the factor graph of SCMA detector and polar soft-output decoder into a joint factor graph, which enables the exchange of messages between SCMA detector and polar decoder iteratively. Simulation results demonstrate that the JIDD receiver has better BER performance and lower complexity than the separate scheme. Specifically, when polar code length $N=256$ and code rate $R = 1/2$ , JIDD outperforms the separate scheme 4.8 and 6 dB over AWGN channel and Rayleigh fading channel, respectively. It also shows that, under 150% system loading, the JIDD receiver only has 0.3 dB performance loss compared to the single user uplink PC-SCMA over AWGN channel and 0.6 dB performance loss over Rayleigh fading channel.

We propose a new class of partially information coupled (PIC) polar codes to improve the transmission efficiency of transport block (TB)-based communication standards. In the proposed PIC polar codes, every two consecutive systematic polar code blocks (CBs) in a TB are coupled by sharing a few systematic information bits. Dummy bits are inserted at the two ends of the TB to construct terminated PIC polar codes. We propose a CB decoding scheme which only uses the information associated with correctly decoded coupled bits to mitigate the serious error propagation problem in successive cancellation based polar code decoding algorithms. We also propose an inter-CB decoding scheme which realizes a windowed decoder with variable window size to achieve a flexible tradeoff between the decoding performance and complexity. We derive a closed form expression for the TB error rate (TBER) of the PIC polar codes. We further optimize the coupling scheme based on the derived TBER. Simulation results confirm the effectiveness of the TBER analysis and the coupling scheme optimization results. They also show that the PIC polar codes can significantly outperform the uncoupled polar codes for various code rates with a slightly increased decoding complexity.

This letter proposes a novel joint decoding scheme for correlated sources using systematic polar codes. In the proposed scheme, each source independently encodes its message into a systematic polar code word and sends the codeword via a binary-input additive white Gaussian noise channel to the common destination, where adaptive cyclic redundancy check-aided successive cancellation list (CA-SCL) decoders are employed. During the iterative decoding process, the log likelihood ratios (LLRs) fed to each adaptive CA-SCL decoder are iteratively combined with the extrinsic LLRs from other decoders. The correlation among the sources is exploited to improve the decoding performance. Simulation results show that, in comparison with independent decoding, the proposed scheme can achieve significantly improved decoding performance without increased complexity.

As an extension of source polarization and channel polarization, this paper considers joint source-channel polarization, which results in a joint source-channel coding (JSCC) scheme using a quasi-uniform systematic polar code (SPC). In this JSCC scheme, the source with side information is encoded as a systematic polar codeword and only parity bits are transmitted through the channel. The indices of systematic bits are quasi-uniform, which enable the source and the channel to be jointly polarized to either a high entropy part or a low entropy part. The analysis reveals that the quasi-uniform SPC cannot be constructed via original polar coding. To solve this problem, additional bit-swap coding is introduced to modify original polar coding and construct this kind of SPCs. The proposed JSCC scheme can asymptotically approach the information-theoretical limit. For the noiseless channel, the proposed scheme is degraded into classic Slepain-Wolf coding or lossless source coding based on parity approach.