Polar codes were introduced by Erdal Arıkan in 2008. They are the first family of errorcorrecting codes that attain the capacity of binaryinput memoryless and symmetric channels with efficient encoding, decoding, and construction algorithms. Since their introduction, polar codes have been generalized and shown to be capacity achieving in numerous other communications settings.
The original construction of polar codes relies on the recursive application of an invertible linear transformation, which, when combined with a successivecancellation decoder, effectively splits the original binaryinput memoryless and symmetric communication channel into a number of bit subchannels. Increasing the recursion depth causes these bit subchannels to converge either to noiseless or purely noisy channels. Virtually errorfree transmission can be achieved by sending the data over noiseless subchannels. While related code constructions had been suggested before (e.g., N. Stolte, I. Dumer and K. Shabunov), Arıkan’s work was the first to prove the polarization phenomenon and thus prove that polar codes are capacity achieving.
Unfortunately, the subchannels converge to these limiting cases relatively slowly, meaning that the errorcorrecting performance of Arıkan’s polar codes improves more slowly with the blocklength than other widelyused codes, such as Turbo and LDPC codes. However, polar codes have been shown to provide excellent errorcorrecting performance with low decoding complexity for practical blocklengths when combined with more advanced decoding algorithms. These favorable traits have led to polar codes being used in the 5G wireless standard, which is a testament to their outstanding performance.
In this Best Readings, we summarize several papers on the theoretical foundations of polarization theory, the construction and decoding of practical polar codes, as well as some generalized polar codes, which can help to overcome limitations of classical Arıkan polar codes. We also focus on practical implementation issues because, despite the simple structure of the encoding and decoding algorithms of polar codes, their practical implementation poses numerous challenges.
Issued June 2019
Contributors
Alexios BalatsoukasStimming, EPFL, Switzerland
Ingmar Land, Huawei Technologies, France
Ido Tal, Technion – Israel Institute of Technology, Israel
Peter Trifonov, Peter the Great St. Petersburg Polytechnic University, Russia
Emanuele Viterbo, Monash University, Australia
Editorial Staff
Matthew C. Valenti
EditorinChief, ComSoc Best Readings
West Virginia University
Morgantown, WV, USA
Topics
After introducing some textbooks, overviews & tutorials, special issues, and standardsrelated articles, we divide the technical papers into the following seven areas:
 Foundations
 Informationtheoretic basics
 Construction and encoding
 Improved and lowcomplexity decoding
 Hardware and software implementation
 Polar coding for general channels
 Generalized polar codes
We note that readers who are mostly interested in the practical aspects of polar coding can focus on topics 1, 3, 4, and 5.
 Books

E. Şaşoğlu, Polarization and Polar Codes, Now Publishers Inc, 2012.
This monograph starts with an explanation of channel polarization and how it is used to construct polar codes. The concept of channel polarization is then generalized to nonbinary input channels and to polarization kernels of sizes larger than the 2x2 kernel used by Arıkan in his seminal work on channel polarization. Finally, some discussion is provided on the joint polarization of multiple random variables with applications to multiuser channels.O. Gazi, Polar Codes: A NonTrivial Approach to Channel Coding, Springer, 2018.
This book explains the philosophy behind the idea of polar encoding from an informationtheoretic perspective. It then discusses the successivecancellation decoding algorithm and associated operations in a tree structure. It also demonstrates the calculation of split channel capacities when polar codes are employed for binary erasure channels, and explains the mathematical formulation of successivecancellation decoding for polar codes.P. Giard, C. Thibeault, and W. J. Gross, HighSpeed Decoders for Polar Codes, Springer International Publishing, 2017.
This book focuses on the implementation of fastSSCbased polar decoders. In particular, the authors consider the hardware and software implementation of standard fastSSC decoding, as well as the hardware implementation of ultrahighspeed unrolled decoders enabled by fastSSC decoding.  Overviews and Tutorials

E. Arıkan, “On the Origin of Polar Coding,” IEEE Journal on Selected Areas in Communications, vol. 34, no. 2, pp. 209223, February 2016.
This paper gives a historical overview of the steps that lead to the discovery of polar codes. Several preexisting techniques and their relation to polar coding are described in detail, giving readers very useful intuition.K. Niu, K. Chen, J. Lin, and Q. T. Zhang, “Polar Codes: Primary Concepts and Practical Decoding Algorithms,” IEEE Communications Magazine, vol. 52, no. 7, pp. 6069, July 2014.
This tutorial paper provides a simple discussion of the main principles behind polar coding. The authors discuss the notion of channel polarization and how it is used in order to construct polar codes. A wide range of decoding algorithms is explained, including successivecancellation decoding, improved and fast successivecancellation decoding algorithms, and beliefpropagation decoding. Finally, the authors provide a comparison of the errorcorrecting performance of polar codes with WCDMA and LTE turbo codes, as well as WiMax LDPC codes.A. BalatsoukasStimming, P. Giard, and A. Burg, “Comparison of Polar Decoders with Existing LowDensity ParityCheck and Turbo Decoders,” in Proc. IEEE Wireless Communications and Networking Conference (WCNC) Workshops, San Francisco, USA, March 2017.
This survey paper compares polar codes with the LDPC codes used in the WiGig, WiFi, and 10GE standards, and the Turbo codes used in the LTE standard. The decoder parameters are selected so that the polar codes match the errorcorrecting performance of the LDPC and turbo codes, and a hardware implementation complexity comparison is performed by scaling the implementation complexity of polar decoders in the literature accordingly.P. Giard, G. Sarkis, A. BalatsoukasStimming, Y. Fan, C.Y. Tsui, A. Burg, C. Thibeault, and W. J. Gross, “Hardware Decoders for Polar Codes: An Overview,” in Proc. IEEE International Symposium on Circuits and Systems (ISCAS), Montreal, Canada, May 2016.
This survey paper provides an overview of hardware implementations of successivecancellation, successivecancellation list, and beliefpropagation decoders for polar codes. The main techniques employed in the literature for each type of decoder are discussed and implementation results for all decoders are summarized and compared.S. Shao, P. Hailes, T.Y. Wang, J.Y. Wu, R. G. Maunder, B. M. AlHashimi, and L. Hanzo, “Survey of Turbo, LDPC and Polar Decoder ASIC Implementations,” IEEE Communications Surveys & Tutorials, early access, January 2019.
This paper is a survey that covers all the codes that were candidates for the 5G standard, including polar codes. The authors first provide a highlevel discussion on the requirements of the 5G standard and the main principles behind each coding scheme. Moreover, ASIC decoders for the various coding schemes are compared in terms of several key characteristics, such as throughput, hardwareefficiency, and errorcorrecting performance. The authors conclude the survey with useful design recommendation as well as some future research directions.  Special Issues

“Recent Advances in Capacity Approaching Codes,” IEEE Journal On Selected Areas In Communications, vol. 34, no. 2, February 2016.
 StandardsRelated Articles

D. Hui, S. Sandberg, Y. Blankenship, M. Andersson, and L. Grosjean, “Channel Coding in 5G New radio: A Tutorial Overview and Performance Comparison with 4G LTE,” IEEE Vehicular Technology Magazine, vol. 13, no. 4, pp. 6069, December 2018.
This tutorial article describes the specific polar and LDPC codes adopted by the 5G NR standard. The purpose of each key component in these codes and the associated operations are explained, and the performance and implementation advantages of these new codes are compared with those of 4G LTE.V. Bioglio, C. Condo, and I. Land, “Design of Polar Codes in 5G New Radio,” arXiv:1804.04389v2, January 2019.
This tutorial provides a description of the encoding chain of polar codes, as specified in the 5G NR standard (3GPP 38.212). While the standard specification provides all details, this tutorial aims at assisting the reader’s comprehension by restructuring the presentation, highlighting the underlying polar coding principles, and relating these principles to the literature.3rd Generation Partnership Project (3GPP), “Multiplexing and Channel Coding,” 3GPP 38.212 V.15.5.0 (201903).
This document is the official technical specification produced by the 3rd Generation Partnership Project (3GPP). It provides the full specification of the polar codes for the 5G NR control channels. The document may be updated and new versions may be made available by the TSG.  Topic: Foundations

E. Arıkan, “Channel Polarization: A Method for Constructing CapacityAchieving Codes for Symmetric BinaryInput Memoryless Channels,” IEEE Transactions on Information Theory, vol. 55, no. 7, pp. 30513073, July 2009.
This is the seminal paper in which polar codes were first introduced. The paper introduces the concept of channel polarization for memoryless channels with binary input. It proves that, if the underlying channel is symmetric, then the code rate tends to the channel capacity while the probability of error tends to zero, assuming the use of an efficient successivecancellation decoder that the paper introduces.E. Arıkan and E. Telatar, “On the Rate of Channel Polarization," in Proc. IEEE International Symposium on Information Theory (ISIT), Seoul, South Korea, June 2009.
This paper considers the same channel polarization setting as Arıkan’s seminal paper, but it contains an improved asymptotic analysis of the probability of incorrect decoding. Specifically, it shows that the error probability is approximately 2^{}^{√}^{N}, where N is the blocklength of the polar code.E. Arıkan, “Source Polarization,” in Proc. IEEE International Symposium on Information Theory (ISIT), Austin, USA, June 2010.
This paper shows how polar codes can be generalized to a source coding setting in order to losslessly compress a binary independent and identically distributed source with side information. Since the realization of the side information is only needed by the decoder, this scheme can be used in a SlepianWolf setting.  Topic: InformationTheoretic Basics

S. B. Korada and R. Urbanke, "Polar Codes are Optimal for Lossy Source Coding," IEEE Transactions on Information Theory, vol. 56, no. 4, pp. 17511768, April 2010.
This paper shows how polar codes can be used for lossy source coding. The coding rate achieved is optimal, provided that the source is symmetric. By incorporating ideas from the fourth paper in this topic, the proposed scheme can also lead to an optimal compression rate for nonsymmetric sources.S. H. Hassani, K. Alishahi, and R. Urbanke, “FiniteLength Scaling of Polar Codes,” IEEE Transactions on Information Theory, vol. 60, no. 10, pp. 58755898, October 2014.
This paper was the first to explore the scaling of polar codes, which is the relation between the code rate R and the blocklength N for a fixed probability of error. Specifically, the paper gives upper and lower bounds for N as a function of the code rate R, the channel capacity I(W), and two constants α and β that depend only on the fixed probability of error and on the transmission channel.D. Goldin and D. Burshtein, “Improved Bounds on the Finite Length Scaling of Polar Codes,” IEEE Transactions on Information Theory, vol. 60, no. 11, pp. 69666978, November 2014.
This paper also deals with the scaling of polar codes. In particular, the authors provide an improved bound on the required blocklength N to communicate reliably at a given rate R. The improved results are also extended to the case of lossy source coding.J. Honda and H. Yamamoto, "Polar Coding without Alphabet Extension for Asymmetric Channels," IEEE Transactions on Information Theory, vol. 59, no. 12, pp. 78297838, December 2012.
This paper shows how polar codes can be used to produce an input distribution that is not symmetric. Thus, it shows a polar coding scheme that extends Arıkan’s original scheme and is capacity achieving for memoryless channels that are not symmetric.V. Guruswami and P. Xia, “Polar Codes: Speed of Polarization and Polynomial Gap to Capacity,” IEEE Transactions on Information Theory, vol. 61, no. 1, pp. 316, January 2015.
This paper shows that, for binaryinput memoryless symmetric channels, the blocklength N of a polar code grows polynomially in the reciprocal of the difference between the code rate and the channel capacity when imposing a particular constraint on the error probability. Moreover, the paper tracks the polarization of channels without resulting to limiting arguments on martingales.E. Şaşoğlu and I. Tal, “Polar Coding for Processes with Memory,” IEEE Transactions on Information Theory, vol. 65, no. 4, pp. 19942003, April 2019.
This paper extends polar codes to a setting in which either the channel or the input distribution (or both) have memory. Slow polarization is proved for both the lowentropy and highentropy sets, while fast polarization is proved only for the lowentropy set.B. Shuval and I. Tal, “Fast Polarization for Processes with Memory,” IEEE Transactions on Information Theory, vol. 65, no. 4, pp. 20042020, April 2019.
This paper closes the gap left by the previous paper. Specifically, it shows that if we assume that the input/output process is governed by an underlying regular and hidden Markov process, then we have fast polarization of both the lowentropy and the highentropy sets. It is also shown that the resulting coding scheme has the same asymptotic probability of error as in the memoryless case, i.e., approximately 2^{}^{√}^{N}.  Topic: Construction and Encoding

I. Tal and A. Vardy, “How to Construct Polar Codes,” IEEE Transactions on Information Theory, vol. 59, no. 10, pp. 65626582, October 2013.
This paper introduces a method for the construction of polar codes. The bit subchannels induced by Arıkan’s polarizing transformation have an output alphabet whose size grows exponentially with the code length. Channel upgrading and degrading transformations are presented, which allow faithful approximations of a subchannel with an intractably large alphabet by another channel having a manageable alphabet size. This enables the construction of polar codes with complexity growing linearly with the blocklength.R. Mori and T. Tanaka, “Performance of Polar Codes with the Construction Using Density Evolution,” IEEE Communications Letters, vol. 13, no. 7, pp. 519521, July 2009.
The authors of this paper were among the first to explicitly state the problem of construction of polar codes. Density evolution was suggested to solve this problem. Furthermore, a partial order is identified on bit subchannels that can be exploited to simplify the construction process.P. Trifonov, “Efficient Design and Decoding of Polar Codes,” IEEE Transactions on Communications, vol. 60, no. 11, pp. 32213227, November 2012.
This paper introduces a method for the construction of polar codes for the AWGN channel. The subchannels induced by Arıkan’s polarizing transformation are approximated with Gaussian ones, and their reliability is characterized by the mean values of the corresponding loglikelihood ratios. An improved decoding algorithm is also presented, which exploits the representation of a polar code as a generalized concatenated code.S. N. Hong, D. Hui, and I. Marić, “CapacityAchieving RateCompatible Polar Codes,” IEEE Transactions on Information Theory, vol. 63, no. 12, pp. 76207632, December 2017.
This work introduces a method of constructing ratecompatible polar codes that are capacityachieving at multiple code rates, and have lowcomplexity decoders. The proposed codes consist of a parallel concatenation of multiple polar codes with an informationbit divider at the input of each polar encoder. It is shown that the proposed codes are capacity achieving for an arbitrary sequence of rates and for any class of degraded channels.G. He, J.C. Belfiore, I. Land, G. Yang, X. Liu, Y. Chen, R. Li, J. Wang, Y. Ge, R. Zhang, and W. Tong, “BetaExpansion: A Theoretical Framework for Fast and Recursive Construction of Polar Codes”, in Proc. IEEE Global Communications Conference (GLOBECOM), Singapore, December 2017.
This paper presents a polarization weight algorithm, which is a simple method for the evaluation of the reliability of polar code subchannels. The authors show that polar codes can be recursively constructed by continuously solving several polynomial equations at each recursive step. The sequences derived from the polarization weight algorithm are shown to satisfy the universal partial order for polar codes.P. Trifonov and V. Miloslavskaya, “Polar Subcodes,” IEEE Journal on Selected Areas in Communications, vol. 34, no. 2, pp. 254266, February 2016.
The paper introduces a generalization of polar codes, where some input symbols of a polarizing transformation, called dynamic frozen or paritycheck frozen symbols, are set to linear combinations of other symbols instead of fixed values. This enables generic linear block codes to be decoded using the techniques developed for polar codes. A method for the construction of subcodes of extended BCH codes is presented, which outperform polar codes with CRC under successivecancellation list decoding.H. Vangala, Y. Hong, and E. Viterbo, “Efficient Algorithms for Systematic Polar Encoding,” IEEE Communications Letters, vol. 20, no. 1, pp. 1720, January 2016.
This paper presents three systematic encoding algorithms for polar codes. These encoders work for any arbitrary choice of frozen bit indices, and they allow a tradeoff between the number of binary computations and the number of bits of memory required by the encoder. The complexity of the best of these encoders is shown to be exactly equal to that of a nonsystematic encoding algorithm.  Topic: Improved and LowComplexity Decoding

I. Tal and A. Vardy, “List Decoding of Polar Codes,” IEEE Transactions on Information Theory, vol. 61, no. 5, pp. 22132226, May 2015.
This paper introduces a successivecancellation list decoder for polar codes, which is a generalization of the classic successivecancellation decoder proposed by Arıkan. Simulations show that the proposed algorithm provides near maximumlikelihood decoding, even for moderate values of the list size. The paper also presents a construction of polar codes with a CRC, which far outperforms classical polar codes under successivecancellation list decoding.K. Chen, K. Niu, and J. Lin, “Improved Successive Cancellation Decoding of Polar Codes,” IEEE Transactions on Communications, vol. 61, no. 8, pp. 31003107, August 2013.
This paper introduces the successivecancellation stack decoding algorithm, which enables reducedcomplexity decoding of polar codes. Moreover, unified descriptions of the successivecancellation, successivecancellation list, and successivecancellation stack decoding algorithms are given as pathsearch procedures on the code tree of polar codes.P. Trifonov, “A Score Function for Sequential Decoding of Polar Codes,” in Proc. IEEE International Symposium on Information Theory (ISIT), Vail, USA, June 2018.
This paper introduces a score function for sequential (stack) decoding of polar codes. A significant reduction of the average decoding complexity is achieved by biasing the path metrics in the minsum version of the stack successivecancellation decoding algorithm with its expected value for the correct path. The proposed approach can also be used for near maximumlikelihood decoding of short extended BCH codes.O. Afisiadis, A. BalatsoukasStimming, and A. Burg, “A lowComplexity Improved Successive Cancellation Decoder for Polar Codes,” in Proc. Asilomar Conference on Signals, Systems and Computers, Pacific Grove, USA, November 2014.
This paper describes successivecancellation flip decoding, where successivecancellation decoding failures are detected using a CRC and additional successivecancellation decoding attempts are made by flipping some bitdecisions. The errorcorrecting performance of successivecancellation flip decoding is significantly improved with respect to successivecancellation decoding with a negligible average complexity overhead, at the cost of a variable running time.M. Rowshan and E. Viterbo, “Stepped List Decoding for Polar Codes,” in Proc. IEEE International Symposium on Turbo Codes & Iterative Information Processing, Hong Kong, December 2018.
This paper investigates the list decoding process by introducing a new parameter named path metric range. The paper proposes a stepwise adaptation of the list size based on the path metric range. This approach preserves the errorcorrection performance of conventional successivecancellation list decoding, while significantly reducing the memory requirements and the computational complexity.A. AlamdarYazdi and F. R. Kschischang, “A Simplified SuccessiveCancellation Decoder for Polar Codes,” IEEE Communications Letters, vol. 15, no. 12, pp. 13781380, December 2011.
This paper describes a modification on the successivecancellation decoder for polar codes, in which local decoders for rateone constituent codes at any depth are simplified. This modification reduces the decoding latency and algorithmic complexity of the conventional SC decoder, while preserving the errorcorrecting performance.S. Cammerer, M. Ebada, A. Elkelesh, S. ten Brink, “Sparse Graphs for Belief Propagation Decoding of Polar Codes,” in Proc. IEEE International Symposium on Information Theory (ISIT), Vail, USA, June 2018.
This paper considers beliefpropagation decoding of polar codes. The authors show how to interpret a polar code as a lowdensity paritycheck (LDPC)like code with an underlying sparse decoding graph. As a result, iterative polar decoding can be conducted on a sparse graph using a fully parallel sumproduct algorithm. The proposed iterative polar decoder has a negligible performance loss for shorttointermediate code lengths compared to Arıkan’s original beliefpropagation decoder, while having lower complexity.  Topic: Hardware and Software Implementation

C. Leroux, A. J. Raymond, G. Sarkis, and W. J. Gross, “A SemiParallel SuccessiveCancellation Decoder for Polar Codes,” IEEE Transactions on Signal Processing, vol. 61, no. 2, pp. 289299, January 2013.
This paper describes a semiparallel hardware architecture for successivecancellation decoding that uses the available hardware resources efficiently. In particular, it is shown how reusing computational logic and memory can significantly reduce the implementation complexity. Most subsequent hardware implementations of decoders based on successive cancellation in the literature use semiparallel architectures.Y. Fan and C.Y. Tsui, “An Efficient PartialSum Network Architecture for SemiParallel Polar Codes Decoder Implementation,” IEEE Transactions on Signal Processing, vol. 62, no. 12, pp., 31653179, June 2014.
This paper describes efficient hardware architectures for the computation of the partial sums in SCbased decoders. The authors explain how a partial sum architecture can be constructed that scales particularly well to large blocklengths, and also how the proposed architecture can be incorporated into a semiparallel SC decoder. FPGA and ASIC results show significant improvements with respect to previous partial sum architectures.G. Sarkis, P. Giard, A. Vardy, C. Thibeault, and W. J. Gross, “Fast Polar Decoders: Algorithm and Implementation,” IEEE Journal on Selected Areas in Communications, vol. 32, no. 5, pp. 946957, May 2014.
This paper presents an improved version of simplified successivecancellation decoding by introducing three new corresponding node types: a singleparitycheckcode node, a repetitioncode node, and a special node whose left child corresponds to a repetition code and its right to a singleparitycheck code. The paper also proposes an algorithm, hardware architecture, and FPGA implementation for the socalled “fastSSC” decoder.A. BalatsoukasStimming, M. Bastani Parizi, and A. Burg, “LLRBased Successive Cancellation List Decoding of Polar Codes,” IEEE Transactions on Signal Processing, vol. 63, no. 19, pp. 51655179, October 2015.
This work reformulates successivecancellation list decoding using loglikelihood ratios (LLRs) by introducing an LLRbased path metric. LLRbased successivecancellation list decoding is equivalent to the original successivecancellation list decoding algorithm, but hardware implementation results show that the LLRbased formulation leads to a significant improvement in the area and operating frequency of successivecancellation list decoders. All subsequent hardware implementations of successivecancellation list decoding in the literature use LLRbased formulation.A. BalatsoukasStimming, M. Bastani Parizi, and A. Burg, “On Metric Sorting for Successive Cancellation List Decoding of Polar Codes,” in Proc. IEEE International Symposium on Circuits and Systems (ISCAS), Lisbon, Portugal, May 2015.
This paper was the first to focus on the critical task of path metric sorting in successivecancellation list decoding. Some properties of the LLRbased path metric are exploited in order to significantly simplify several wellknown sorting architectures. ASIC synthesis results show significant improvements in area and operating frequency with respect to existing sorting architectures.B. Yuan and K. K. Parhi, “Early Stopping Criteria for EnergyEfficient LowLatency BeliefPropagation Polar Code Decoders,” IEEE Transactions on Signal Processing, vol. 62, no. 24, pp. 64966506, December 2014.
This paper describes several heuristic techniques for early stopping in beliefpropagation decoding of polar codes. Each technique works best for a different SNR regime, so an adaptive SNRdependent early stopping method is also presented. Hardware implementation results show that the early stopping techniques can significantly improve the average throughput and energyefficiency of beliefpropagation decoders.B. Le Gal, C. Leroux, and C. Jego, “MultiGb/s Software Decoding of Polar Codes,” IEEE Transactions on Signal Processing, vol. 63, no. 2, pp. 349359, January 2015.
This paper shows how the parallelism capabilities of modern CPUs can be used to implement very highspeed SC decoders in software, which are of great interest in softwaredefined radio applications. It is also explained how existing algorithmic simplifications for hardware SC decoders can be beneficial in software implementations. The implemented software decoders achieve throughputs of more than 1 Gb/s for a wide range of blocklengths and code rates.  Topic: Polar Coding for General Channels

U. U. Fayyaz and J. R. Barry, “Polar Codes for Partial Response Channels,” in Proc. IEEE International Conference on Communications (ICC), Budapest, Hungary, June 2013.
The paper deals with polar codes for partialresponse channels using turbo equalization at the receiver side. The original successivecancellation decoder for polar codes does not produce softoutputs for code bits. The authors propose a softinput softoutput variant of the successivecancellation decoder, called “soft cancellation (SCAN)” decoder, which is suitable for such turbo receiver architectures and keeps the computational complexity low.E. Şaşoğlu, E. Telatar, and E. M. Yeh, “Polar Codes for the TwoUser MultipleAccess Channel,” IEEE Transactions on Information Theory, vol. 59, no. 10, pp. 65836592, October 2013.
This paper extends Arıkan’s polar coding method to twouser multipleaccess channels. Using Arıkan’s construction for each of the two users of the channel results in a coding scheme whose sum rate is the one that corresponds to uniform input distributions. The asymptotic encoding and decoding complexities and the errorcorrecting performance of these codes are shown to be the same as in the singleuser case.N. Goela, E. Abbe, and M. Gastpar, “Polar Codes for Broadcast Channels,” IEEE Transactions on Information Theory, vol. 61, no. 2, pp. 758782, February 2015.
This paper introduces polar codes for discrete memoryless broadcast channels. For muser deterministic broadcast channels, the polarizationbased codes are shown to achieve rates on the boundary of the privatemessage capacity region. For twouser noisy broadcast channels, polar implementations are presented for two informationtheoretic schemes, namely Cover’s superposition codes and Marton’s codes.M. Mondelli, S. H. Hassani, I. Sason, and R. Urbanke, “Achieving Marton's Region for Broadcast Channels Using Polar Codes,” IEEE Transactions on Information Theory, vol. 61, no. 2, pp. 783800, February 2015.
This paper presents polar coding schemes for the twouser discrete memoryless broadcast channel, which achieve Marton’s region with both common and private messages. This is the best achievable rate region known to date, and it is tight for all classes of twouser discrete memoryless broadcast channels whose capacity regions are known.M. Seidl, A. Schenk, C. Stierstorfer, and J. B. Huber, “PolarCoded Modulation,” IEEE Transactions on Communications, vol. 61, no. 10, pp. 41084119, October 2013.
A framework is proposed that allows for a joint description and optimization of binary polar coding and pulseamplitude modulation schemes. Multilevel coding and bitinterleaved coded modulation are considered, the conceptual equivalence of polar coding and multilevel coding is detailed, and rules for the optimum choice of the labelling are developed.D. Zhou, K. Niu, and C. Dong, “Construction of Polar Codes in Rayleigh Fading Channel,” IEEE Communications Letters, vol. 23, no. 3, pp. 402405, March 2019.
This paper addresses the construction of polar codes for the Rayleigh fading channel. Two algorithms are proposed to determine an AWGN channel that is equivalent to the actual Rayleigh fading channel. For this equivalent AWGN channel, the frozen set is determined using density evolution under a Gaussian approximation.  Topic: Generalized Polar Codes

S. B. Korada, E. Şaşoğlu, and R. Urbanke, “Polar Codes: Characterization of Exponent, Bounds, and Constructions,” IEEE Transactions on Information Theory, vol. 56, no. 12, pp. 62536264, December 2010.
This paper shows that any ℓ × ℓ matrix with the property that none of its column permutations is upper triangular polarizes binaryinput memoryless channels. Then, it characterizes the exponent of a given square matrix and provides upper and lower bounds on the achievable exponents. Using these bounds, the authors show that there are no matrices of size smaller than 15×15 with exponents exceeding 1/2. Furthermore, a general construction based on BCH codes which achieves exponents arbitrarily close to 1 for large ℓ is given. Specifically, for size 16×16, this construction yields an exponent greater than 1/2.R. Mori and T. Tanaka, “Source and Channel Polarization Over Finite Fields and Reed–Solomon Matrices,” IEEE Transactions on Information Theory, vol. 60, no. 5, pp. 27202736, May 2014.
This paper generalizes Arıkan’s channel polarization for the binary field alphabet to polarization over any finite field, with the field size q being a power of a prime. A necessary and sufficient condition for a matrix over a finite field is shown under which any source and channel are polarized. Additionally, the polarization speed result for binary alphabets is generalized to arbitrary finite fields. This paper also provides an explicit construction of an ℓ×ℓ matrix, based on a ReedSolomon matrix, with the asymptotically fastest polarization for ℓ ≤ q.A. Fazeli, H. Hassani, M. Mondelli, and A. Vardy, “Binary Linear Codes with Optimal Scaling: Polar Codes with Large Kernels,” in Proc. IEEE Information Theory Workshop (ITW), Guangzhou, China, November 2018.
This paper shows that for the binary erasure channel, there exist ℓ × ℓ binary kernels with quasilinear encoding and decoding complexity, such that polar codes constructed from these kernels achieve capacity with a scaling exponent that tends to the optimal value of 2 as ℓ grows.V. Bioglio, I. Land, F. Gabry, and J.C. Belfiore, “Flexible Design of MultiKernel Polar Codes by Reliability and Distance Properties,” in Proc. International Symposium on Turbo Codes & Iterative Information Processing (ISTC), Hong Kong, December 2018.
This paper proposes a polar code construction for multikernel polar codes by taking into account both reliability and distance. The new design provides a framework to optimize the code design for successivecancellation list decoding, allowing performance to trade against decoding complexity.N. Presman, O. Shapira, S. Litsyn, T. Etzion, and A. Vardy, “Binary Polarization Kernels from Code Decompositions,” IEEE Transactions on Information Theory, vol. 61, no. 5, pp. 2227–2239, May 2015.
This paper proposes designing binary kernels with a large exponent by using code decompositions. The proposed kernels are generally nonlinear, but they provide a better polarization exponent than the previously known kernels of the same dimensions. In particular, nonlinear kernels of dimensions 14, 15, and 16 are constructed and are shown to have optimal asymptotic errorcorrecting performance.G. Trofimiuk and P. Trifonov, “Efficient Decoding of Polar Codes with Some 16×16 Kernels,” in Proc. IEEE Inf. Theory Workshop (ITW), Guangzhou, China, November 2018.
This paper presents reduced complexity decoding algorithms for 16×16 polarization kernels with polarization rate 0.51828 and scaling exponents 3.346 and 3.450. It shows that, with these kernels, increasing the list size in the successivecancellation list decoder provides a significant performance gain compared to the case of Arıkan’s 2×2 kernel. The proposed approach results in lower decoding complexity compared to polar decoding with Arıkan’s kernel at the same performance level.