PAC Codes Meet CRC-Polar Codes

Jinhong Yuan; Mohammad Rowshan; Xinyi Gu

REVIEW 4 major objections 5 minor 1 cited by

PAC Codes Meet CRC-Polar Codes

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Placing parity-determined bits at the most reliable coordinates destroys most minimum-weight codewords, buying up to 0.5 dB for profile-shifted PAC codes and 0.12 dB for continuous CRC-polar codes.

desk verdict Genuinely new code constructions with real but modest gains; the MWC enumeration explaining them is unproven, so the paper deserves review rather than rejection. read the letter →

arxiv 2501.18080 v2 pith:ARBR64ZS submitted 2025-01-30 cs.IT math.IT

Xinyi Gu , Mohammad Rowshan , Jinhong Yuan This is my paper · ORCID

classification cs.ITmath.IT

keywords polarcodesPACCRC-polarminimumweightcodewordserrorcoefficientlistdecodingrateprofiledesignpre-transformation

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

CRC-polar codes owe part of their error-rate edge to an accidental side effect: the CRC parity bits land on the most reliable (largest-index) coordinates, where they carry message-forced values that break the row combinations which would otherwise form minimum-weight codewords. The paper makes this mechanism deliberate, proposing profile-shifted PAC (PS-PAC) codes that reserve a block of large-index frozen coordinates in the rate profile, and shows that this removes about 94.6% of the minimum-weight codewords of the parent PAC code for a (64,32) code while gaining up to 0.5 dB in block error rate under list decoding. A second proposal, continuous CRC-polar (CCRC-polar) codes, spreads the running remainders of CRC division over frozen coordinates and gains 0.12 dB over CRC-polar codes at length 512. If the mechanism is as stated, weight-distribution design, not just decoder improvements, is a direct lever on the error floor of short polar-type codes.

What carries the argument

The engine is the coset decomposition of polar codewords: every nonzero codeword is a leading row $g_i$ of the polar transform plus a combination of later rows, and the formation theorem [17] fixes which later rows can appear in a minimum-weight combination, namely a subset $J$ of rows whose support meets the leader in exactly one extra bit, plus a derived set $M(J)$. A minimum-weight codeword survives only if those rows can be chosen freely, so a coordinate whose value is predetermined (a no-freedom bit) that lies inside $J \cup M(J)$ breaks the combination. Condition (8) is the paper's cancellation rule for such parity-determined coordinates, stated without proof of sufficiency or necessity. The two proposed schemes are two ways of manufacturing these coordinates: PS-PAC codes reserve large-index frozen coordinates at the end of the rate profile and precode them with the PAC convolution, so the frozen set intersects $(i, N-1]$ and the first condition of Lemma 1 fails; CCRC-polar codes mask frozen coordinates with running CRC remainders that the list decoder tracks bit by bit.

What would settle it

Encode every message word of a short code, for example the (32,16) case of Example 1 or the (64,32) design of Table I, over $\mathbb{F}_2$, collect all codewords at the true minimum distance, and compare the true coset-by-coset survival with the prediction of condition (8). Any codeword predicted to be canceled that still exists, or one predicted to survive that is missing, shows the condition is not exact, and the enumeration underlying the claimed gain would be unreliable; the same exhaustive count also supplies the true $A_{w_{\min}}$ to check against Table I.

Watch

Extended reading notes

Core claim

The paper's central claim is that the error-rate advantage of CRC-polar codes under list decoding is largely explained by an accident of placement: parity bits occupy the most reliable coordinates, carry a value forced by the message rather than a free choice, and when such a forced coordinate lands in the row sets $J \cup M(J)$ needed to build a minimum-weight codeword, the combination is destroyed. Condition (8) states the cancellation rule, and the paper turns the accident into a design rule. PS-PAC codes freeze the $\alpha$ most reliable coordinates and let the PAC precoder turn them into no-freedom bits, cutting the error coefficient $A_{w_{\min}}$ from 504 to 27 for a (64,32) code (94.6% reduction) and from 320 to 27 for (64,48), with up to 0.5 dB measured gain at length 64. Example 1 shows the mechanism concretely: the polar codeword $g_{24}+g_{25}$ has weight 4, but with the forced CRC-bit pattern the same leading row produces weight 20. CCRC-polar codes instead mask frozen coordinates with the running remainder of CRC division, gaining 0.12 dB over CRC-polar codes at length 512.

Load-bearing premise

The count of surviving minimum-weight codewords rests on equation (8), the claim that a parity bit cancels a minimum-weight codeword exactly when it takes value 1 outside $J \cup M(J)$ and 0 inside, and the paper does not prove this condition is both necessary and sufficient; if it is not exact, the enumerated $A_{w_{\min}}$ values and the link to the 0.5 dB gain would need revision.

Editorial extensions

If this is right

PS-PAC codes eliminate roughly 94.6% of the minimum-weight codewords of the parent PAC code for (64,32) and about 91.6% for (64,48), shrinking the error coefficient that the union bound blames for the error floor.
Under list decoding with list size 32, PS-PAC codes gain up to 0.5 dB over both PAC and CRC-polar codes at length 64, with the gain tapering at high SNR where the steeper CRC-polar slope closes the gap.
For longer codes (N = 256 and 512), profile shifting lets PS-PAC codes recover the suboptimal regime of plain PAC codes and match CRC-polar codes in block error rate.
CCRC-polar codes outperform standard CRC-polar codes by 0.12 dB at length 512, and match them at length 64, where too few frozen coordinates are available for remainder masking.
If the parity constraints eliminate every minimum-weight codeword of the underlying polar code, the minimum distance of the concatenated code increases, lifting the whole union-bound curve.

Reading between the lines

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

A design rule the paper leaves implicit: the fixed shift $\alpha$ could be replaced by a combinatorial search over rate profiles that cover the union of $J \cup M(J)$ over the dominant cosets, potentially pushing $A_{w_{\min}}$ below the residual 27 achieved here.
Because CCRC decoding stores one running remainder per list path, memory grows with the number of masked frozen coordinates; a sliding-window remainder variant might retain most of the 0.12 dB gain at lower cost, a testable engineering variant.
The gains are measured at list size 32; under smaller lists, where type I errors dominate, the weight-distribution reduction should matter less and path-metric effects more, so the empirical ranking of PS-PAC versus CRC-polar codes could differ in that regime.
Condition (8), if exact, implies an ordering principle for parity placement: put the CRC bits precisely on the coordinates that appear in $J \cup M(J)$ of low-weight cosets, which could make the cancellation complete rather than partial.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: this is a genuine engineering contribution with two new code constructions and plausible, consistently reproduced BLER gains. The theoretical explanation for the gains—the MWC enumeration based on Eq. (8)—is asserted rather than proved, so treat Table I as heuristic until the authors close that gap.

What's new: PS-PAC codes reserve the α most reliable coordinates as frozen, precoded by the convolution, which breaks up many minimum-weight row combinations. That's a simple, direct idea, and the simulations show it buys up to 0.5 dB for N=64 and 0.1–0.2 dB for longer codes versus PAC and CRC-polar. CCRC-polar codes continuously insert intermediate CRC remainders into frozen positions, and that gives a small but consistent extra 0.12 dB at N=512. Both constructions are easy to implement from the description. The paper also provides a coset-based breakdown of how CRC bits interact with MWC formation, which is a useful way to think about concatenated polar codes.

Where it's soft: the enumeration in Table I rests entirely on condition (8), which says a MWC is canceled if the CRC bits take specific values at coordinates in J∪M(J). That condition is plausible as a per-codeword cancellation test, but the paper never shows it is sufficient and necessary for the whole codebook. New MWCs can form from row combinations that were not MWCs in the underlying polar code, and combinations involving multiple CRC constraints could interact. So the A_wmin values in Table I are unverified. No proof, no enumeration script, no error bars, and no plain-polar baseline makes it harder to separate the effect of the rate-profile shift from the precoding. The "0.1-2 dB" should be 0.1-0.2 dB.

None of this kills the paper. The performance claims are empirical and the code constructions are well defined. If the enumeration is eventually shown to be incomplete, the designs still stand on their simulation results; only the MWC-based explanation would need to be softened.

Who this is for: people working on short-blocklength codes, PAC codes, CRC-polar concatenation, and list decoding. It deserves a serious referee; the proof gap in Section IV is exactly what review should probe. I'd send it out, with a request that the authors either prove the enumeration or relocate Table I into a heuristic remark.

Referee Report

4 major / 5 minor

Summary. This paper studies minimum-weight codewords (MWCs) of polar, PAC, CRC-polar, and newly proposed variants. The authors propose two schemes: profile-shifted PAC (PS-PAC) codes, which reserve α no-freedom coordinates at the large-index end of the rate profile, and continuous CRC-polar (CCRC-polar) codes, which replace frozen bits by intermediate remainders of the CRC division. The central analytical claim is that a condition stated in Eq. (8) determines, for each CRC bit coordinate, whether an MWC of the underlying polar code is eliminated. Using this condition, the paper reports reduced MWC counts in Table I and claims BLER gains of up to 0.5 dB for short PS-PAC codes and an additional 0.12 dB for CCRC-polar codes at N=512. Simulations with list decoding (L=32) are presented for N=64, 256, 512.

Significance. If the MWC-elimination mechanism in Eq. (8) is rigorously established and the enumeration of remaining MWCs is correct, the paper offers a useful design principle: reserving no-freedom large-index coordinates can reduce the error coefficient of PAC and CRC-polar codes. The PS-PAC construction is simple, and the idea of using intermediate CRC remainders in frozen positions is interesting. The paper also builds on a substantial body of prior work on MWC formation in polar/PAC codes. However, the manuscript does not supply a formal proof of Eq. (8), does not give an enumeration procedure or code for Table I, and the simulation section lacks error bars and a plain-polar baseline. Because the 0.5 dB and 0.12 dB gains are explained through the reduced MWC counts, these gaps are load-bearing for the paper's main claims.

major comments (4)

[Section IV, Eq. (8)] The cancellation condition in Eq. (8) is stated without proof of sufficiency or necessity, and the text only says that MWCs are canceled 'if' the condition holds. This condition tests an individual row combination of the original polar code, but the CRC-polar, PS-PAC, and CCRC-polar codes are new linear codes with constrained input spaces. New row combinations that did not form MWCs in the original polar code could form MWCs, and multiple CRC constraints could interact. The paper does not show that each surviving original MWC is the only possible MWC, nor does it provide an enumeration algorithm with a correctness proof. Since Table I and the claimed performance gains are attributed to the reduced Awmin values, Eq. (8) must be proved or replaced by a verifiable codebook-level enumeration.
[Table I and Fig. 4] For the (64,32) and (64,48) codes, Table I reports Awmin values of 6 and 13 for CRC-polar but 63 and 306 for CCRC-polar, a factor of roughly 10 to 23 more MWCs. Yet Fig. 4 states that CRC-polar and CCRC-polar have identical error-correction performance for both rates. If the MWC count is the main explanatory quantity for the error floor, this large discrepancy should produce a visible difference in the high-SNR region. The manuscript offers only a brief explanation about limited frozen coordinates for short codes, which does not address why a 10-50x change in the error coefficient has no effect. This inconsistency needs to be resolved, either by correcting the enumeration, by explaining why MWCs do not dominate in this regime, or by reporting results separately.
[Section V-B, CCRC-polar encoding and decoding] The encoding of CCRC-polar codes is under-specified. Expressions such as 'r = [i0, ..., i, 0] / q(x)' and 'r = [ij-t, ij] / q(x)' are not defined as vector-valued operations, and the truncation and cyclic repetition rules for the remainders are only described informally. The list-decoder modification is also not described in enough detail: the text says the decoder 'performs CRC decoding bit by bit' and 'stores and updates remainders,' but no path-metric update rule or complexity analysis is given. This makes the CCRC-polar construction and its decoder non-reproducible from the manuscript.
[Section VI, Numerical Results] The performance claims of up to 0.5 dB and 0.12 dB are point estimates without error bars, confidence intervals, or a statement of the number of simulated blocks. The curves are not accompanied by a plain polar baseline, and only one CRC polynomial and one convolutional polynomial are used. Given that the 0.12 dB CCRC-polar gain is small, simulation uncertainty could alter the conclusion. The authors should provide confidence intervals or at least the number of trials, and should clarify whether the reported gains are stable across code constructions and decoder settings.

minor comments (5)

[Section VI, paragraph after Fig. 5] The text says 'an overall power gain of 0.1-2 dB'; this is presumably a typo for 0.1-0.2 dB. Correcting this would avoid ambiguity.
[Section IV, paragraph after Example 1] The phrase 'elimination of WMCs' should read 'elimination of MWCs'.
[Equation (5) and following text] The word 'extened' is a typo and should read 'extended'.
[Example 1] The notation IPolar and ICRC is used without a formal definition in the example; the sets are described inline, but a short definition would improve readability.
[Section IV, first paragraph] The statement that CRC bits occupy 'the t most reliable bit coordinates... corresponding to the largest-index coordinates in the rate profile' conflates reliability order with index order. The mapping should be stated explicitly, especially since Eq. (8) depends on the positions of R relative to J and M(J).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's central claims rest on prior analytic lemmas and are validated by independent simulations, with no fitted parameter renamed as a prediction.

full rationale

The paper's derivation chain is not circular. The key mechanism—reserving large-index coordinates so that no-freedom bits block the row combinations in (5)—is an application of Lemma 1 in [7] and the minimum-weight codeword (MWC) formation theorem in [17], both of which are analytic results about the polar transform that do not assume the present paper's conclusions. The CRC cancellation condition in Eq. (8) is a direct per-codeword consistency check: for a row combination with support {i} ∪ J ∪ M(J), a CRC coordinate r must equal 1 if r is in that support and 0 otherwise; Eq. (8) simply tests whether the CRC-determined bit matches the combination. Because CRC-polar codewords are a subset of the underlying polar code, testing the original MWCs is a principled enumeration route, and Table I's A_wmin values are not fitted to the BLER curves. The reported gains are obtained from list-decoding simulations at fixed list size, independent of the weight-counting step. The main correctness risk—that for PS-PAC codes the Table I enumeration may miss newly formed same-weight combinations because nonzero frozen rows extend beyond the underlying polar code—is a proof gap, not a circularity. Self-citations to [7], [16], and [17] are real, parameter-free prior results and are not used to redefine the target quantity. No equation in the paper reduces to its own input, and no fitted parameter is presented as a prediction.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The central claims rest on the unproven condition in Eq. (8) and on standard assumptions about the polar transform and the channel model. The design parameters alpha and p are chosen by the authors, not derived, and the results are sensitive to them.

free parameters (2)

alpha (number of shifted bit coordinates) = 8
The number of large-index coordinates moved from the information set to the frozen set is chosen as alpha=8 in all PS-PAC simulations. The performance gain depends on this choice, and no sensitivity analysis is provided.
PAC convolutional polynomial p = [1 0 1 1 0 1 1 0 1 1]
A specific convolutional polynomial is used for all PAC and PS-PAC simulations. It is adopted from prior work, but the scheme's performance relies on it; no optimization or justification for this specific choice is given.

assumptions (3)

standard math The characterization of minimum-weight codeword formation in polar and PAC codes from [17, Theorem 1] is correct, including the definitions of Ki and M(J).
The paper uses this theorem as the basis for its analysis of MWC formation and reduction.
domain assumption The BI-AWGN channel and SC list decoding with list size L=32 model the error performance faithfully.
All numerical results are obtained under this model, stated in Section VI.
ad hoc to paper The condition in Eq. (8) exactly determines whether a minimum-weight codeword is eliminated by a parity bit at a CRC coordinate.
Eq. (8) is asserted in Section IV without a formal proof. The enumeration of MWCs in CRC-polar and PS-PAC codes depends on this condition being both necessary and sufficient.

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Cite this review

Pith. "Pith review of PAC Codes Meet CRC-Polar Codes." pith.science (2026). https://pith.science/paper/ARBR64ZS

@misc{pith2026250118080,
  author       = {Pith},
  title        = {Pith review of: PAC Codes Meet CRC-Polar Codes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ARBR64ZS}},
  note         = {Machine review of arXiv:2501.18080}
}

read the original abstract

CRC-Polar codes under SC list decoding are well-regarded for their competitive error performance. This paper examines these codes by focusing on minimum weight codewords, breaking them down into the rows of the polar transform. Inspired by the significant impact of parity check bits and their positions, we apply a shifted rate-profile for polarization-adjusted convolutional (PS-PAC) codes, thereby achieving similar improvements in the weight distribution of polar codes through precoding. The results demonstrate a significant improvement in error performance, achieving up to a 0.5 dB power gain with short PS-PAC codes. Additionally, leveraging convolutional precoding in PAC codes, we adopt a continuous deployment (masking) of parity check bits derived from the remainder of continuous division of the partial message polynomial and the CRC polynomial over frozen positions in the rate-profile. This approach enhances performance for medium-length codes, with an overall improvement of 0.12 dB.

Figures

Figures reproduced from arXiv: 2501.18080 by the authors.

**Figure 1.** Precoding and encoding of polar codes. is the minimum weight, may significantly decrease relative to polar codes (without pre-transformation). Hence, from the union bound [18, Sect. 10.1], we expect that this reduction potentially improves the block error rate (BLER) of a (near) maximum likelihood decoding for a binary input additive white Gaussian noise (BI-AWGN) channel. In the conventional PAC coding, forward con… view at source ↗

**Figure 3.** Construction of CCRC-polar codes in natural order. For coordinates i0 ≤ i < it, i ∈ I, the data sequence is insufficient to compute the first intermediate remainder. To resolve this, a zero sequence 0 is appended to the data sequence [i0, i] to reach the required length t + 1 and r = [i0, · · · , i, 0] / q(x). For coordinates i ≥ it, i ∈ I, r = [ij−t, ij ] / q(x), where t < j ≤ K. The bits at coordinates in D are re… view at source ↗

**Figure 5.** Performance of codes with N = 256, 512. In [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Performance of (64, 32) and (64, 48) codes. [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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Towards Weight Distribution-Aware Polar Codes
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A rule-based swap of a few information positions between adjacent degree levels improves minimum distance and reported block error rate for 5G-based polar and PAC codes.

Reference graph

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

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work page 2015

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arXiv 1908

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M. Rowshan and E. Viterbo, “On convolutional precoding i n pac codes,” in 2021 IEEE Globecom W orkshops (GC Wkshps) , 2021, pp. 1–6

work page 2021

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M. Rowshan, A. Burg, and E. Viterbo, “Polarization-adju sted convo- lutional (pac) codes: Fano decoding vs list decoding,” arXiv preprint arXiv:2002.06805, 2020

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H. Y ao, A. Fazeli, and A. V ardy, “List decoding of arıkan’ s pac codes,” Entropy, vol. 23, no. 7, p. 841, 2021

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M. Rowshan and J. Y uan, “On the minimum weight codewords o f pac codes: The impact of pre-transformation,” IEEE Journal on Selected Areas in Information Theory , vol. 4, pp. 487–498, 2023

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