Pith. sign in

REVIEW 3 major objections 5 minor 10 cited by

From sequential decoding to channel polarization and back again

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read PAC codes, a convolution followed by the polar transform and sequential decoding, bring finite-blocklength performance close to the random-coding dispersion bound while retaining polar codes' capacity-achieving property.

desk verdict The PAC code construction is a genuinely new idea with a sound capacity claim, but the near-dispersion FER result is a single unreproduced simulation and is honestly labeled as unproven. read the letter →

arxiv 1908.09594 v3 pith:H63H6KYU submitted 2019-08-26 cs.IT math.IT

Erdal Ar{i}kan This is my paper · ORCID
classification cs.ITmath.IT MSC 94B1094B3594A24
keywords channelpolarizationpolarcodesPACsequentialdecodingFanodecodercutoffratefiniteblocklengthBIAWGN
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper retells the intellectual path from sequential decoding to channel polarization, then uses that path to introduce a new code family, polarization-adjusted convolutional (PAC) codes. The central claim is that placing an outer convolutional code before the polar transform and decoding the resulting irregular tree code with a Fano sequential decoder yields codes that still achieve channel capacity—they contain polar codes as the special case where the convolution is the identity—and that at practical blocklengths they perform far better than polar codes. The paper's headline simulation shows a PAC code with $N=128$, rate $1/2$, and a weight-ordered (Reed-Muller) rate profile achieving frame error rates close to the dispersion approximation for the binary-input additive-white-Gaussian-noise (BIAWGN) channel, the finite-blocklength benchmark for random codes. A sympathetic reader should care because this offers a concrete route to closing the gap between polar coding's asymptotic optimality and its short-blocklength performance.

What carries the argument

The central object is the PAC encoding transform $x = v T P_n$, where $v$ is a rate-profiled data carrier, $T$ is an upper-triangular Toeplitz matrix implementing convolution with impulse response $c$, and $P_n = [[1,0],[1,1]]^{\otimes n}$ is the polar transform. Decoding uses a Fano sequential (depth-first tree-search) decoder over the irregular tree code generated by $T$ under the constraint that frozen coordinates are zero, with a time-varying bias metric computed recursively as in successive cancellation. This construction is an upper-lower decomposition of a generator matrix: it separates coding into a sparse convolution and a fast polar transform, and the paper argues that for good choices of the data index set $A$ and $c$, the combined matrix $G = T P_n$ looks sufficiently random to give near-dispersion performance while keeping encoding complexity $O(N \log N)$.

What would settle it

Simulate PAC codes with Reed-Muller rate profiling on the binary-input additive-white-Gaussian-noise channel at $N=256$ and $N=512$, rate $1/2$, and compare their frame error rates against the dispersion approximation at those lengths; if the gap widens as $N$ grows, or if per-block decoding complexity grows super-polynomially even though the rate profile stays below the polarized cutoff-rate profile, the "sufficiently random" heuristic and the complexity guide would be refuted.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the cutoff-rate boosting that motivated polar coding can be recovered at finite blocklengths by undoing the 0-1 rate simplification of polar codes: instead of freezing bit-channels, PAC codes run a convolutional code over the polarized bit-channels and decode the whole system as one irregular tree code. The paper reports that with the Reed-Muller design rule for the data index set and a suitably chosen convolution, the overall transform $G = T P_n$ behaves as if it were a random code, bringing the frame error rate at $N=128$, $R=1/2$ close to the binary-input additive-white-Gaussian-noise (BIAWGN) dispersion approximation for error rates above $10^{-3}$. It further claims that PAC codes achieve channel capacity in general because an identity convolution reduces them to polar codes.

Load-bearing premise

The load-bearing premise is that at the blocklengths of interest the combined transform $G = T P_n$ behaves statistically like a random code, so PAC codes inherit the near-maximum-likelihood performance predicted by the dispersion approximation; the paper presents this as an informal heuristic, not a proof.

Editorial extensions

If this is right

  • PAC codes achieve channel capacity on symmetric binary-input memoryless channels, since taking the convolution to be the identity recovers polar codes.
  • At finite blocklengths PAC codes can outperform polar codes under both successive-cancellation and CRC-aided successive-cancellation list decoding, as the $N=128$, rate-$1/2$ BIAWGN simulation shows.
  • The rate-profile heuristic gives a design rule: a data index set whose cumulative rate stays below the polarized cutoff-rate profile should keep Fano decoding complexity manageable at that signal-to-noise ratio.
  • The best simulated performance came from a weight-ordered (Reed-Muller) rate profile, suggesting PAC codes may tolerate channel parameter variations better than polar codes; the paper leaves a rigorous universal-design statement open.
  • The main practical obstacle is the variable complexity of sequential decoding, and the paper points to fixed-complexity alternatives such as list Viterbi and beam search as future directions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the "sufficiently random" explanation is correct, PAC codes should track the dispersion approximation across a range of rates and blocklengths; testing this at $N=256$ and $512$ and at rates away from $1/2$ would turn the heuristic into a measurable prediction.
  • The condition that the rate profile stay below the polarized cutoff-rate profile resembles a finite-length error-exponent comparison; making it precise could connect sequential-decoding complexity to coding error exponents rather than only to capacity.
  • The upper-lower-decomposition viewpoint suggests searching for other sparse factorizations of generator matrices: any fast transform paired with a compatible outer trellis might yield codes with near-dispersion behavior under an appropriate decoder.
  • If the Reed-Muller rate profile is shown to be universal across binary-input memoryless channels of a given capacity, PAC codes would become channel-agnostic finite-length codes, a stronger property than the channel-specific rate profiles used for polar codes.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The manuscript is a written version of the 2019 Shannon Lecture. It traces the conceptual path from sequential decoding, through Massey's cut-off-rate example and Pinsker's scheme, to multi-level coding and then to polar coding. It then introduces polarization-adjusted convolutional (PAC) codes, which place an outer convolutional transform T before the inner polar transform P_n, and reports by computer simulation that an N=128, R=1/2 PAC code has frame error rate close to the BIAWGN dispersion approximation (Fig. 12). The paper further argues that PAC codes contain polar codes as a special case and hence can achieve channel capacity. Theorems 1 and 2 are standard polarization results with proofs cited to the literature; the new PAC-code performance claims rest on a single simulation and an informal 'looks sufficiently random' heuristic, and Section VIII explicitly acknowledges that performance and complexity of PAC codes are not yet rigorously studied.

Significance. If substantiated, the finite-length PAC-code claim would be practically notable: a rate-1/2 code of blocklength 128 with FER near the finite-blocklength dispersion bound would markedly improve on the polar and CA-SCL curves shown in Fig. 12, while using sequential decoding. The capacity-achieving statement via the polar-code special case (T equal to the identity and A a polar data index set) is correct and does not depend on any simulation. The paper also provides a useful historical synthesis of the ideas that led to polar coding, with correct citations to the polarization literature. However, the central quantitative claim is currently supported only by one unreproduced simulation, and the paper itself states that PAC-code performance and complexity remain open. The significance is therefore conditional on the reproducibility and further validation of that simulation.

major comments (3)
  1. [Section VII, Fig. 12] The paper's principal quantitative claim—that a PAC code with N=128, R=1/2, RM design rule, and c=(1,0,1,1,0,1,1) has FER near the BIAWGN dispersion approximation—is supported by exactly one simulation. The decoder description in Section VII is incomplete: only 'time-varying bias' is mentioned, with no specification of the metric, threshold update policy, search limits, or stopping rule, and Fig. 12 provides no confidence intervals, repetition counts, or measured complexity statistics. The claim is therefore not reproducible from the manuscript. Please either give full decoder pseudocode and complete simulation data, or explicitly label the curve as preliminary simulation evidence rather than a demonstrated performance result.
  2. [Section VII, paragraph beginning 'Evidently'] The explanation for the near-dispersion behavior—that the combined transform G = TP_n 'looks sufficiently random'—is an informal heuristic. No random-like property is defined or measured; for example, the paper does not report the weight enumerator, distance profile, or comparison with random-code ML decoding. Since this heuristic is the only principle offered to connect the simulation to the dispersion bound, it needs to be either formulated as a testable conjecture with supporting analysis or replaced by a quantitative evaluation of the claimed 'sufficiently random' property.
  3. [Section VIII, second paragraph] The manuscript explicitly states that 'the performance and complexity of PAC codes are yet to be studied rigorously' and that understanding the computational complexity of the sequential decoder is an open problem. This is in tension with the motivational use in Section VII of the rate-profile criterion—that staying below the polarized cutoff rate profile indicates low Fano-decoder complexity—as a practical design guide. Without complexity statistics from the reported simulation or an analysis of Fano search effort, the complexity side of the PAC-code proposal is unverified. The paper should clearly separate this open heuristic from the rigorous polar-code results in Theorem 2.
minor comments (5)
  1. [Abstract] The phrase 'original idea s' contains a stray space; please proofread for similar typographical errors.
  2. [Section VII, paragraph after Fig. 13] 'an codeword u' should be 'a codeword u'.
  3. [Section VII, last paragraph] The claim that the RM design rule 'suggests that, unlike polar codes, PAC codes are robust against channel parameter variations' is not supported by the single simulation at one SNR setting; please rephrase as a conjecture or add supporting experiments across channel parameters.
  4. [Section VI, Fig. 12 caption] The dispersion approximation is described as 'an estimate of the average ML-decoding performance' of a random code ensemble; in the cited reference [19] the normal approximation is a rate approximation for the maximal achievable rate, not an ensemble average. The wording should be adjusted for precision.
  5. [Section VII, paragraph on design rules] The observation that the Fano decoder 'ran significantly faster' under the polar design rule is reported without any measured complexity data; consider adding mean or median search effort, or a histogram of decoder complexity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the PAC capacity claim follows by containment of polar codes, and the finite-length FER claim is an explicitly reported simulation, not a derived prediction.

full rationale

The paper's central asymptotic claim is that PAC codes achieve channel capacity because they contain polar codes as a special case: Section VII states the cutoff-rate score function 'recovers polar codes when T is set to the identity transform (corresponding to c = 1)', and Section VIII concludes that 'in general PAC codes can achieve channel capacity since they contain polar codes as a special case.' This is a valid inclusion argument, not a circular one. The only supporting citation for polar-code capacity achievability is Arikan's 2009 theorem, an external, parameter-free mathematical result whose assumptions do not include PAC performance. The finite-blocklength FER near the dispersion approximation is presented as an observed simulation result — 'Fig. 12 presents the result of a computer simulation with a PAC code with N = 128, R = 1/2, A chosen in accordance with the RM design rule' — and is not claimed to be a first-principles derivation. The informal explanation that the combined transform 'looks sufficiently random' is a heuristic, but invoking a heuristic to explain a simulation is not circular. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the authors' prior work, and no definition is loaded with the target conclusion. Accordingly, the paper warrants a circularity score of 0.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The ledger separates the established polarization machinery from the heuristic PAC proposal. The free parameters are design choices in the single simulation. The axioms include standard cited theorems and the informal assumptions needed for the new PAC performance claims. No invented physical entities appear.

free parameters (2)
  • convolution impulse response c in simulation = (1,0,1,1,0,1,1)
    Chosen by hand for the N=128, R=1/2 BIAWGN example. The paper states that A matters more than c and that c could be random when the constraint length is large, but no systematic search or derivation is given.
  • data index set A via RM design rule = N=128, K=64, s(i)=w(i-1), top K scores
    The RM score function is selected as the best among the tested rules (capacity, cutoff rate, RM) for FER performance. The paper says RM was the best design found, but this is an empirical choice, not a theorem.
assumptions (6)
  • standard math Channel polarization theorem for transforms Pn (Theorem 1, [13]): the fraction of bit-channel capacities near 1 tends to C(W).
    Invoked to justify that 0-1 rate assignments yield capacity-achieving polar codes. Proved in Arikan 2009; not re-derived here.
  • standard math Polar code FER bound Pe <= sum_{i in A} Z(W_i) and SC decoding runtime O(N log N) (Theorem 2, [13], [17]).
    Used to claim that polar codes achieve capacity with low complexity. Accepted as established results from the cited literature.
  • domain assumption Idealized successive-decoding bit-channels Wi: Ui -> (Y, U^{i-1}) with no decision errors are sufficient for deriving polar codes.
    Section V states that for purposes of deriving polar codes it suffices to consider the ideal case with no decision errors. This is a modeling idealization.
  • domain assumption Symmetric BMC restriction for polar and PAC coding.
    Section VI restricts polar coding to symmetric binary-input memoryless channels. The capacity and FER statements are scoped to that class.
  • ad hoc to paper The combined transform G = TPn behaves like a random code at finite blocklengths.
    Section VII asserts informally that G 'looks sufficiently random' to explain near-dispersion FER. No proof or characterization is provided.
  • ad hoc to paper Fano sequential decoding with a time-varying bias has acceptable complexity when the rate profile stays below the polarized cutoff rate profile.
    Section VII proposes this as a heuristic design guide. Section VIII explicitly lists a better understanding of the sequential decoder complexity as an open problem.

how reviews work

0 comments
Cite this review

Pith. "Pith review of From sequential decoding to channel polarization and back again." pith.science (2026). https://pith.science/paper/H63H6KYU

@misc{pith2026190809594,
  author       = {Pith},
  title        = {Pith review of: From sequential decoding to channel polarization and back again},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H63H6KYU}},
  note         = {Machine review of arXiv:1908.09594}
}
read the original abstract

This note is a written and extended version of the Shannon Lecture I gave at 2019 International Symposium on Information Theory. It gives an account of the original ideas that motivated the development of polar coding and discusses some new ideas for exploiting channel polarization more effectively in order to improve the performance of polar codes.

Figures

Figures reproduced from arXiv: 1908.09594 by the authors.

Figure 2
Figure 2. Order of main topics discussed in the note. [PITH_FULL_IMAGE:figures/full_fig_p001_2.png] view at source ↗
Figure 1
Figure 1. Channel coding system. Shannon [1] proved that for a broad class of channels, there exists a channel parameter C, called capacity, such that arbitrarily reliable transmission (small Pe) is attainable at any given rate R if R < C (and unattainable if R > C). Shannon’s theorem settled the question about the trade-off between the rate ( R) and reliability ( Pe) in a communication system. However, the random-coding anal… view at source ↗
Figure 4
Figure 4. Tree representation of a convolutional code. [PITH_FULL_IMAGE:figures/full_fig_p002_4.png] view at source ↗
Figures from the paper (12 more)
Figure 5
Figure 5. Figure 5: Ratio of cutoff rate to capacity for the BSC. [PITH_FULL_IMAGE:figures/full_fig_p003_5.png]
Figure 6
Figure 6. Figure 6: Pinsker’s scheme. d1 CE u1 uˆ1 SD dˆ1 d2 CE u2 uˆ2 SD dˆ2 dK CE uK uˆK SD dˆK b b b b b b b b b b b b b b b b b b b b b W1 W2 WK [PITH_FULL_IMAGE:figures/full_fig_p004_6.png]
Figure 7
Figure 7. Figure 7: Bit-channels created by Pinsker’s scheme. [PITH_FULL_IMAGE:figures/full_fig_p004_7.png]
Figure 8
Figure 8. Figure 8: The mapper in the figure is a one-to-one transformatio [PITH_FULL_IMAGE:figures/full_fig_p004_8.png]
Figure 8
Figure 8. Figure 8: Multi-level coding d1 CE1 u1 y SD1 ˆd1 d2 CE2 u2 yuˆ 1 SD2 ˆd2 dN CEN uN yuˆ N−1 SDN ˆdN b b b b b b b b b b b b b b b b b b b b b W1 W2 WN [PITH_FULL_IMAGE:figures/full_fig_p005_8.png]
Figure 9
Figure 9. Figure 9: Bit channels created by MLC/MSD correct. For purposes of deriving polar codes, it suffices to consider only this ideal case with no decision errors. Hence, from now on, we suppose that Wi has this ideal form. An important property of the MLC/MSD scheme is the conservat…
Figure 11
Figure 11. Figure 11: Capacity and cutoff rate profiles over BIAWGN channe [PITH_FULL_IMAGE:figures/full_fig_p006_11.png]
Figure 10
Figure 10. Figure 10: Channel polarization for BIAWGN channel at 3 dB SNR. [PITH_FULL_IMAGE:figures/full_fig_p006_10.png]
Figure 12
Figure 12. Figure 12: Performance curves over the BIAWGN channel. [PITH_FULL_IMAGE:figures/full_fig_p007_12.png]
Figure 13
Figure 13. Figure 13: As with polar codes, the natural block lengths for PAC codes are powers of two, N = 2n , n ≥ 1. The code dimension K can be any integer between 1 and N. The encoding operation for PAC codes is as follows. A rate-profiling block inserts the source word d into a data ca…
Figure 13
Figure 13. Figure 13: PAC coding scheme. polar transform. A low-complexity encoding alternative is to compute first u = vT and then x = uPn. As usual, we characterize the convolution operation by an impulse response c = (c0, · · · , cm), where by convention we assume that c0 6= 0 and cm 6=…
Figure 14
Figure 14. Figure 14: Irregular tree code example. To summarize, a PAC code is specified by four parameters (N, K, A, c). In simulation studies we observed that the per￾formance of a PAC code is more sensitive to the choice of A than to c. As long as the constraint length of the convolutio…

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 10 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. BiD Codes: Algebraic Codes from $3 \times 3$ Kernel

    cs.IT 2025-07 conditional novelty 7.0 of 10

    BiD codes, built from a 3x3 Kronecker kernel, have proven minimum distance growing at least as N^0.543 at any fixed rate, faster than Reed-Muller's N^0.5.

  2. Lower Bounds for Error Coefficients of Griesmer Optimal Linear Codes via Iteration

    cs.IT 2025-07 conditional novelty 6.0 of 10

    The authors derive five iterative lower bounds on error coefficients of Griesmer-optimal linear codes, prove tightness for most binary cases up to dimension 5, and determine all 5-dimensional AFER-optimal binary codes.

  3. Low-Complexity Decoding for Low-Rate Block Codes of Short Length Based on Concatenated Coding Structure

    cs.IT 2025-05 conditional novelty 6.0 of 10

    A concatenated-coding construction with an improved MRIP ordering from inner-code soft outputs reaches near-ML block error rates for short low-rate codes at lower A* decoding complexity.

  4. Rate-Matching Deep Polar Codes via Polar Coded Extension

    cs.IT 2025-05 conditional novelty 6.0 of 10

    Extended deep polar codes concatenate the intermediate polar codewords of each pre-transform layer to achieve non-power-of-two blocklengths with a soft-output SCL decoder, outperforming conventional rate-matching at m...

  5. Subcode Ensemble Decoding of Polar Codes

    cs.IT 2025-04 conditional novelty 6.0 of 10

    A new type of polar-code pre-transformation enables subcode ensemble decoding, which matches the error performance of larger list-size decoders at the same hardware list budget.

  6. PAC Codes Meet CRC-Polar Codes

    cs.IT 2025-01 conditional novelty 6.0 of 10

    Reserving high-index coordinates as frozen bits in PAC codes and embedding intermediate CRC remainders into frozen positions improves error-correction performance by up to 0.5 dB for short codes and 0.12 dB for longer codes.

  7. PAC codes with Bounded-Complexity Sequential Decoding: Pareto Distribution and Code Design

    cs.IT 2024-12 conditional novelty 6.0 of 10

    Proves that PAC codes with rate profiles below polarized cutoff rates have Pareto-bounded sequential decoding complexity, and demonstrates a code built on this rule that gains over 0.75 dB at FER 1e-5.

  8. The Optimization of Random Tree Codes for Limited Computational Resources

    cs.IT 2025-01 conditional novelty 5.0 of 10

    A finite-computation achievability bound for sequential decoding of random tree codes, optimized by a greedy bit-placement rule, approaches the random-coding union and Gallager benchmarks for pure random codes.

  9. DeepPolar+: Breaking the BER-BLER Trade-off with Self-Attention and SMART (SNR-MAtched Redundancy Technique) decoding

    cs.IT 2025-06 reject novelty 4.0 of 10

    An attention-based neural polar decoder with a block-level loss improves simulated BER and BLER for a (256,37) polar code, and a CRC-guided ensemble variant reports further gains.

  10. Polarized Element-pair Code Based FFMA over a Gaussian Multiple-access Channel

    cs.IT 2025-06 conditional novelty 4.0 of 10

    A polarization-based finite-field multiple-access code with two decoders is shown by simulation to beat polar random spreading by about 1.25 dB for 15 users on a Gaussian multiple-access channel.

Reference graph

Works this paper leans on

25 extracted references · 24 canonical work pages · cited by 10 Pith papers

  1. [1]

    A mathematical theory of communication,

    C. E. Shannon, “A mathematical theory of communication, ” The Bell System Technical Journal , vol. 27, pp. 379–423, July 1948

    work page 1948

  2. [2]

    Coding for noisy channels,

    Peter Elias, “Coding for noisy channels,” in IRE Convention Record, Part 4, pp. 37–46, Mar. 1955

    work page 1955

  3. [3]

    Sequential Decoding for Reliable Com munication,

    J. M. Wozencraft, “Sequential Decoding for Reliable Com munication,” Tech. Report 325, Res. Lab. Elect., M.I.T., Aug. 1957

    work page 1957

  4. [4]

    R. G. Gallager, Information Theory and Reliable Communication . New Y ork: Wiley, 1968

    1968

  5. [5]

    J. M. Wozencraft and I. M. Jacobs, Principles of Communication Engi- neering. New Y ork: Wiley, 1965

    work page 1965

  6. [6]

    Sequential Decoding for Multiple Access Cha nnels,

    E. Arıkan, “Sequential Decoding for Multiple Access Cha nnels,” Tech. Rep. LIDS-TH-1517, Lab. Inf. Dec. Syst., M.I.T., 1985

    work page 1985

  7. [7]

    A perspective on multiaccess channels,

    R. Gallager, “A perspective on multiaccess channels,” IEEE Transactions on Information Theory , vol. 31, pp. 124–142, Mar. 1985

    work page 1985

  8. [8]

    On the complexity of decoding,

    M. S. Pinsker, “On the complexity of decoding,” Problemy Peredachi Informatsii, vol. 1, no. 1, pp. 84–86, 1965

    work page 1965

Show all 25 references
  1. [9]

    Capacity, cutoff rate, and coding for a direc t-detection optical channel,

    J. Massey, “Capacity, cutoff rate, and coding for a direc t-detection optical channel,” IEEE Transactions on Communications, vol. 29, pp. 1615–1621, Nov. 1981

    1981

  2. [10]

    On the origin of polar coding,

    E. Arıkan, “On the origin of polar coding,” IEEE Journal on Selected Areas in Communications , vol. 34, pp. 209–223, Feb. 2016

    2016

  3. [11]

    Error-free coding,

    P . Elias, “Error-free coding,” Transactions of the IRE Professional Group on Information Theory , vol. 4, pp. 29–37, Sept. 1954

    1954

  4. [12]

    A new multilevel coding method using error- correcting codes,

    H. Imai and S. Hirakawa, “A new multilevel coding method using error- correcting codes,” IEEE Transactions on Information Theory , vol. 23, pp. 371–377, May 1977

    1977

  5. [13]

    Channel polarization: A method for constru cting capacity- achieving codes for symmetric binary-input memoryless cha nnels,

    E. Arıkan, “Channel polarization: A method for constru cting capacity- achieving codes for symmetric binary-input memoryless cha nnels,” IEEE Transactions on Information Theory , vol. 55, pp. 3051–3073, July 2009

    2009

  6. [14]

    Performance of polar codes with t he con- struction using density evolution,

    R. Mori and T. Tanaka, “Performance of polar codes with t he con- struction using density evolution,” IEEE Communications Letters, vol. 13, pp. 519–521, July 2009

    2009

  7. [15]

    On t he construction of polar codes,

    R. Pedarsani, S. H. Hassani, I. Tal, and E. Telatar, “On t he construction of polar codes,” in 2011 IEEE International Symposium on Information Theory Proceedings, pp. 11–15, IEEE, July 2011

    2011

  8. [16]

    How to construct polar codes,

    I. Tal and A. V ardy, “How to construct polar codes,” IEEE Transactions on Information Theory , vol. 59, pp. 6562–6582, Oct. 2013

    2013

  9. [17]

    On the rate of channel polariz ation,

    E. Arıkan and E. Telatar, “On the rate of channel polariz ation,” in 2009 IEEE International Symposium on Information Theory Pr oceedings, pp. 1493–1495, IEEE, June 2009

    2009

  10. [18]

    Polar codes: speed of polariza tion and polynomial gap to capacity,

    V . Guruswami and P . Xia, “Polar codes: speed of polariza tion and polynomial gap to capacity,” IEEE Transactions on Information Theory , vol. 61, pp. 3–16, Jan. 2015

    2015

  11. [19]

    Channel coding r ate in the finite blocklength regime,

    Y . Polyanskiy, H. Poor, and S. V erd´ u, “Channel coding r ate in the finite blocklength regime,” IEEE Transactions on Information Theory , vol. 56, pp. 2307–2359, May 2010

    2010

  12. [20]

    List decoding of polar codes,

    I. Tal and A. V ardy, “List decoding of polar codes,” in 2011 IEEE International Symposium on Information Theory Proceeding s, pp. 1–5, July 2011

    2011

  13. [21]

    A heuristic discussion of probabilistic deco ding,

    R. Fano, “A heuristic discussion of probabilistic deco ding,” IEEE Transactions on Information Theory , vol. 9, pp. 64–74, Apr. 1963

    1963

  14. [22]

    A class of multiple-error-correcting codes a nd the decoding scheme,

    I. Reed, “A class of multiple-error-correcting codes a nd the decoding scheme,” Transactions of the IRE Professional Group on Information Theory, vol. 4, pp. 38–49, Sept. 1954

    1954

  15. [23]

    Application of Boolean algebra to switch ing circuit design and to error detection,

    D. E. Muller, “Application of Boolean algebra to switch ing circuit design and to error detection,” Transactions of the I.R.E. Professional Group on Electronic Computers, vol. EC-3, pp. 6–12, Sept. 1954

    1954

  16. [24]

    List Viterbi decodin g algorithms with applications,

    N. Seshadri and C. E. W. Sundberg, “List Viterbi decodin g algorithms with applications,” IEEE Transactions on Communications , vol. 42, pp. 313–323, Feb. 1994

    1994

  17. [25]

    Russell and P

    S. Russell and P . Norvig, Artificial Intelligence: A Modern Approach . Upper Saddle River, NJ, USA: Prentice Hall Press, 3rd ed., 20 09

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.