[Ground-station] Polar Codes - join the fun! - recommended reading

[Michelle Thompson]

You're very welcome.

I am wondering if Polar Codes would be useful for our uplink channels?

I know of them from 5G cellular, but I'm not far along enough in the
reading to be able to discern all their strengths.

-Michelle W5NYV




On Wed, Jul 1, 2020 at 8:19 AM Ahmet Inan <xdsopl at gmail.com> wrote:

> Thank you Michelle for all the pointers and readings!
>
> Ahmet
>
> On Tue, Jun 30, 2020 at 10:36 PM Michelle Thompson via Ground-Station
> <ground-station at lists.openresearch.institute> wrote:
> >
> > Here are some recommendations for reading about Polar Codes.
> >
> > Ahmet Inan has started some work here. Please see
> https://github.com/xdsopl/polar
> >
> > Please feel free to join the discussion, ask questions, and make some
> progress.
> >
> > Ahmet has requested a copy of "A Simplified Successive-Cancellation
> Decoder for Polar Codes" by Amin Alamdar Yazdi and Frank R. Kschischang. If
> you have access to this paper and can legally provide a copy to Ahmet, then
> please let me know. Otherwise I will purchase it for him at $15 individual
> member price.
> >
> > I can and will loan him the readings from Transactions on Information
> Theory in the list below.
> >
> > From an IEEE perspective, here are the recommended readings in this area.
> >
> > Books:
> >
> > P. Giard, C. Thibeault, and W. J. Gross, High-Speed Decoders for Polar
> Codes, Springer International Publishing, 2017.
> > This book focuses on the implementation of fast-SSC-based polar
> decoders. In particular, the authors consider the hardware and software
> implementation of standard fast-SSC decoding, as well as the hardware
> implementation of ultra-high-speed unrolled decoders enabled by fast-SSC
> decoding.
> >
> > 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 non-binary 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 multi-user channels.
> >
> > O. Gazi, Polar Codes: A Non-Trivial Approach to Channel Coding,
> Springer, 2018.
> > This book explains the philosophy behind the idea of polar encoding from
> an information-theoretic perspective. It then discusses the
> successive-cancellation 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 successive-cancellation decoding
> for polar codes.
> >
> > Overviews and Tutorials:
> >
> > E. Arıkan, “On the Origin of Polar Coding,” IEEE Journal on Selected
> Areas in Communications, vol. 34, no. 2, pp. 209-223, February 2016.
> > This paper gives a historical overview of the steps that lead to the
> discovery of polar codes. Several pre-existing 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. 60-69, 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 successive-cancellation
> decoding, improved and fast successive-cancellation decoding algorithms,
> and belief-propagation decoding. Finally, the authors provide a comparison
> of the error-correcting performance of polar codes with WCDMA and LTE turbo
> codes, as well as WiMax LDPC codes.
> >
> > A. Balatsoukas-Stimming, P. Giard, and A. Burg, “Comparison of Polar
> Decoders with Existing Low-Density Parity-Check 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, Wi-Fi, and 10GE standards, and the Turbo codes used in the LTE
> standard. The decoder parameters are selected so that the polar codes match
> the error-correcting 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. Balatsoukas-Stimming, 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
> successive-cancellation, successive-cancellation list, and
> belief-propagation 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.
> Al-Hashimi, 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
> high-level 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, hardware-efficiency, and
> error-correcting performance. The authors conclude the survey with useful
> design recommendations as well as some future research directions.
> >
> > Standards-Related
> >
> > 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. 60-69, 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 (2019-03).
> > 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.
> >
> > Foundations of Polar Codes
> >
> > E. Arıkan, “Channel Polarization: A Method for Constructing
> Capacity-Achieving Codes for Symmetric Binary-Input Memoryless Channels,”
> IEEE Transactions on Information Theory, vol. 55, no. 7, pp. 3051-3073,
> 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
> successive-cancellation 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 Slepian-Wolf setting.
> >
> > Information Theory Basis
> >
> > S. B. Korada and R. Urbanke, "Polar Codes are Optimal for Lossy Source
> Coding," IEEE Transactions on Information Theory, vol. 56, no. 4, pp.
> 1751-1768, 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 non-symmetric
> sources.
> >
> > S. H. Hassani, K. Alishahi, and R. Urbanke, “Finite-Length Scaling of
> Polar Codes,” IEEE Transactions on Information Theory, vol. 60, no. 10, pp.
> 5875-5898, 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. 6966-6978, 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. 7829-7838, 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. 3-16, January 2015.
> > This paper shows that, for binary-input 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. 1994-2003, 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 low-entropy and high-entropy sets, while fast
> polarization is proved only for the low-entropy set.
> >
> > B. Shuval and I. Tal, “Fast Polarization for Processes with Memory,”
> IEEE Transactions on Information Theory, vol. 65, no. 4, pp. 2004-2020,
> 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 low-entropy and the high-entropy 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.
> >
> > Construction and Encoding
> >
> > I. Tal and A. Vardy, “How to Construct Polar Codes,” IEEE Transactions
> on Information Theory, vol. 59, no. 10, pp. 6562-6582, 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.
> 519-521, 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. 3221-3227, 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 log-likelihood
> 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, “Capacity-Achieving Rate-Compatible
> Polar Codes,” IEEE Transactions on Information Theory, vol. 63, no. 12, pp.
> 7620-7632, December 2017.
> > This work introduces a method of constructing rate-compatible polar
> codes that are capacity-achieving at multiple code rates, and have
> low-complexity decoders. The proposed codes consist of a parallel
> concatenation of multiple polar codes with an information-bit 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, “Beta-Expansion: 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. 254-266, February
> 2016.
> > The paper introduces a generalization of polar codes, where some input
> symbols of a polarizing transformation, called dynamic frozen or
> parity-check 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 successive-cancellation list decoding.
> >
> > H. Vangala, Y. Hong, and E. Viterbo, “Efficient Algorithms for
> Systematic Polar Encoding,” IEEE Communications Letters, vol. 20, no. 1,
> pp. 17-20, 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
> non-systematic encoding algorithm.
> >
> > Improved and Low-Complexity Decoding
> >
> > I. Tal and A. Vardy, “List Decoding of Polar Codes,” IEEE Transactions
> on Information Theory, vol. 61, no. 5, pp. 2213-2226, May 2015.
> > This paper introduces a successive-cancellation list decoder for polar
> codes, which is a generalization of the classic successive-cancellation
> decoder proposed by Arıkan. Simulations show that the proposed algorithm
> provides near maximum-likelihood 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
> successive-cancellation 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.
> 3100-3107, August 2013.
> > This paper introduces the successive-cancellation stack decoding
> algorithm, which enables reduced-complexity decoding of polar codes.
> Moreover, unified descriptions of the successive-cancellation,
> successive-cancellation list, and successive-cancellation stack decoding
> algorithms are given as path-search 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 min-sum version of the stack
> successive-cancellation decoding algorithm with its expected value for the
> correct path. The proposed approach can also be used for near
> maximum-likelihood decoding of short extended BCH codes.
> >
> > O. Afisiadis, A. Balatsoukas-Stimming, and A. Burg, “A low-Complexity
> Improved Successive Cancellation Decoder for Polar Codes,” in Proc.
> Asilomar Conference on Signals, Systems and Computers, Pacific Grove, USA,
> November 2014.
> > This paper describes successive-cancellation flip decoding, where
> successive-cancellation decoding failures are detected using a CRC and
> additional successive-cancellation decoding attempts are made by flipping
> some bit-decisions. The error-correcting performance of
> successive-cancellation flip decoding is significantly improved with
> respect to successive-cancellation 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 error-correction performance of conventional successive-cancellation
> list decoding, while significantly reducing the memory requirements and the
> computational complexity.
> >
> > A. Alamdar-Yazdi and F. R. Kschischang, “A Simplified
> Successive-Cancellation Decoder for Polar Codes,” IEEE Communications
> Letters, vol. 15, no. 12, pp. 1378-1380, December 2011.
> > This paper describes a modification on the successive-cancellation
> decoder for polar codes, in which local decoders for rate-one constituent
> codes at any depth are simplified. This modification reduces the decoding
> latency and algorithmic complexity of the conventional SC decoder, while
> preserving the error-correcting 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 belief-propagation decoding of polar codes. The
> authors show how to interpret a polar code as a low-density parity-check
> (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 sum-product algorithm. The proposed iterative polar decoder has a
> negligible performance loss for short-to-intermediate code lengths compared
> to Arıkan’s original belief-propagation decoder, while having lower
> complexity.
> >
> > Hardware and Software Implementation
> >
> > C. Leroux, A. J. Raymond, G. Sarkis, and W. J. Gross, “A Semi-Parallel
> Successive-Cancellation Decoder for Polar Codes,” IEEE Transactions on
> Signal Processing, vol. 61, no. 2, pp. 289-299, January 2013.
> > This paper describes a semi-parallel hardware architecture for
> successive-cancellation decoding that uses the available hardware resources
> efficiently. In particular, it is shown how re-using computational logic
> and memory can significantly reduce the implementation complexity. Most
> subsequent hardware implementations of decoders based on successive
> cancellation in the literature use semi-parallel architectures.
> >
> > Y. Fan and C.-Y. Tsui, “An Efficient Partial-Sum Network Architecture
> for Semi-Parallel Polar Codes Decoder Implementation,” IEEE Transactions on
> Signal Processing, vol. 62, no. 12, pp., 3165-3179, June 2014.
> > This paper describes efficient hardware architectures for the
> computation of the partial sums in SC-based 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 semi-parallel 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. 946-957, May 2014.
> > This paper presents an improved version of simplified
> successive-cancellation decoding by introducing three new corresponding
> node types: a single-parity-check-code node, a repetition-code node, and a
> special node whose left child corresponds to a repetition code and its
> right to a single-parity-check code. The paper also proposes an algorithm,
> hardware architecture, and FPGA implementation for the so-called “fast-SSC”
> decoder.
> >
> > A. Balatsoukas-Stimming, M. Bastani Parizi, and A. Burg, “LLR-Based
> Successive Cancellation List Decoding of Polar Codes,” IEEE Transactions on
> Signal Processing, vol. 63, no. 19, pp. 5165-5179, October 2015.
> > This work re-formulates successive-cancellation list decoding using
> log-likelihood ratios (LLRs) by introducing an LLR-based path metric.
> LLR-based successive-cancellation list decoding is equivalent to the
> original successive-cancellation list decoding algorithm, but hardware
> implementation results show that the LLR-based formulation leads to a
> significant improvement in the area and operating frequency of
> successive-cancellation list decoders. All subsequent hardware
> implementations of successive-cancellation list decoding in the literature
> use LLR-based formulation.
> >
> > A. Balatsoukas-Stimming, 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 successive-cancellation list decoding. Some properties of the
> LLR-based path metric are exploited in order to significantly simplify
> several well-known 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 Energy-Efficient
> Low-Latency Belief-Propagation Polar Code Decoders,” IEEE Transactions on
> Signal Processing, vol. 62, no. 24, pp. 6496-6506, December 2014.
> > This paper describes several heuristic techniques for early stopping in
> belief-propagation decoding of polar codes. Each technique works best for a
> different SNR regime, so an adaptive SNR-dependent early stopping method is
> also presented. Hardware implementation results show that the early
> stopping techniques can significantly improve the average throughput and
> energy-efficiency of belief-propagation decoders.
> >
> > B. Le Gal, C. Leroux, and C. Jego, “Multi-Gb/s Software Decoding of
> Polar Codes,” IEEE Transactions on Signal Processing, vol. 63, no. 2, pp.
> 349-359, January 2015.
> > This paper shows how the parallelism capabilities of modern CPUs can be
> used to implement very high-speed SC decoders in software, which are of
> great interest in software-defined 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.
> >
> > 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 partial-response channels using
> turbo equalization at the receiver side. The original
> successive-cancellation decoder for polar codes does not produce
> soft-outputs for code bits. The authors propose a soft-input soft-output
> variant of the successive-cancellation 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 Two-User
> Multiple-Access Channel,” IEEE Transactions on Information Theory, vol. 59,
> no. 10, pp. 6583-6592, October 2013.
> > This paper extends Arıkan’s polar coding method to two-user
> multiple-access 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 error-correcting performance of these
> codes are shown to be the same as in the single-user case.
> >
> > N. Goela, E. Abbe, and M. Gastpar, “Polar Codes for Broadcast
> Channels,”  IEEE Transactions on Information Theory, vol. 61, no. 2, pp.
> 758-782, February 2015.
> > This paper introduces polar codes for discrete memoryless broadcast
> channels. For m-user deterministic broadcast channels, the
> polarization-based codes are shown to achieve rates on the boundary of the
> private-message capacity region. For two-user noisy broadcast channels,
> polar implementations are presented for two information-theoretic 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. 783-800, February
> 2015.
> > This paper presents polar coding schemes for the two-user 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 two-user discrete memoryless
> broadcast channels whose capacity regions are known.
> >
> > M. Seidl, A. Schenk, C. Stierstorfer, and J. B. Huber, “Polar-Coded
> Modulation,” IEEE Transactions on Communications, vol. 61, no. 10, pp.
> 4108-4119, October 2013.
> > A framework is proposed that allows for a joint description and
> optimization of binary polar coding and pulse-amplitude modulation schemes.
> Multilevel coding and bit-interleaved 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. 402-405,
> 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.
> >
> > 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. 6253-6264, December 2010.
> > This paper shows that any ℓ × ℓ matrix with the property that none of
> its column permutations is upper triangular polarizes binary-input
> 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. 2720-2736, 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 Reed-Solomon 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 quasi-linear 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
> Multi-Kernel 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 multi-kernel polar
> codes by taking into account both reliability and distance. The new design
> provides a framework to optimize the code design for
> successive-cancellation 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 error-correcting 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 successive-cancellation 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.
> >
> > -Michelle W5NYV
> >
> > "Potestatem obscuri lateris nescis."
> >
>
>
> --
>
> Ahmet Inan
>
> Co-founder and CEO of aicodix GmbH
> https://www.aicodix.de/
>
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