{"id":"6c60d367-77dc-45c0-821a-195ad88e9a98","arxiv_id":"2501.18080","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"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.","lead":"This paper introduces two modifications to PAC and CRC-polar codes: shifting the rate profile to place frozen bits at high-reliability positions, and embedding intermediate CRC remainders into frozen coordinates. The first modification claims up to 0.5 dB gain for short codes, the second about 0.12 dB gain for longer codes.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (8) is a cancellation test for individual polar MWCs, not a codebook enumeration; newly formed same-weight combinations may be missed, so Table I's A_wmin values are unverified.","rationale":"The reader's weakest assumption points to Eq. (8), and that is exactly where the paper's theoretical load is concentrated. The claimed gains are explained by a reduction in minimum-weight codewords, and Table I is the only quantitative evidence connecting the construction to the BLER curves. Eq. (8) gives a plausible condition for destroying an individual minimum-weight row combination, but the paper never proves that the CRC-polar or PS-PAC codebook cannot contain new minimum-weight codewords formed by different combinations, especially combinations that include the CRC/frozen rows themselves. Without that proof, Table I may be a lower bound on the true error coefficient, and the attribution of the simulated gains to MWC reduction remains conditional. The proposed test—brute-force or algorithmic weight enumeration of the actual code generator matrices for the shortest cases—directly settles whether the enumeration is correct. If the values match, the central theoretical mechanism is supported for those cases; if they do not, the explanatory claim needs revision. Since the reader already assigned CONDITIONAL with moderate confidence, this stress-test does not change the verdict, but it sharpens the condition under which the paper's central claim would be accepted.","tokens_in":9559,"tokens_out":4171,"duration_ms":46844,"concrete_test":"Take the exact generator matrices of the (64,32) and (64,48) CRC-polar and PS-PAC codes used in Table I (same rate profiles, CRC polynomial q(x)=x^11+x^10+x^9+x^5+1, PAC polynomial p=[1 0 1 1 0 1 1 0 1 1], α=8). Compute the true minimum distance and the number of minimum-weight codewords with a standard linear-code weight-enumeration algorithm (e.g., Brouwer-Zimmermann or Canteaut-Chabaud, as implemented in Magma or SageMath), without invoking Eq. (8). If the resulting A_wmin equals Table I for both codes, Eq. (8)-based enumeration is validated for these cases; if not, the MWC-reduction argument and the BLER explanation are not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section IV uses condition (8) to decide when a minimum-weight codeword of the underlying polar code is eliminated by CRC bits. The condition is plausible as a per-codeword cancellation test: if a CRC coordinate that must participate in the row combination is forced to 0, or an extra row is forced to 1, that particular combination is destroyed. What is not established is that this individual test enumerates the MWC count of the resulting CRC-polar or PS-PAC code. The codebook is a new linear code: inputs are constrained at R (or F) by CRC/convolution relations, so row combinations that did not form MWCs in the original polar code may now form MWCs, and combinations involving multiple CRC bit constraints may interact. The paper gives no proof that each surviving original MWC is the only possible MWC, nor that Eq. (8) is sufficient as well as necessary for cancellation. Since Table I and the claimed 0.5 dB / 0.12 dB gains are explained through reduced A_wmin, an unverified enumeration makes the central explanatory claim conditional. The article itself does not supply the proof of (8) or the codebook-level enumeration, and no code or enumeration script is provided.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":9846,"tokens_out":5536,"duration_ms":64550,"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":[{"comment":"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.","section":"Section IV, Eq. (8)"},{"comment":"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":"Table I and Fig. 4"},{"comment":"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":"Section V-B, CCRC-polar encoding and decoding"},{"comment":"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.","section":"Section VI, Numerical Results"}],"minor_comments":[{"comment":"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":"Section VI, paragraph after Fig. 5"},{"comment":"The phrase 'elimination of WMCs' should read 'elimination of MWCs'.","section":"Section IV, paragraph after Example 1"},{"comment":"The word 'extened' is a typo and should read 'extended'.","section":"Equation (5) and following text"},{"comment":"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":"Example 1"},{"comment":"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).","section":"Section IV, first paragraph"}],"recommendation":"major_revision","confidential_remarks":"The paper relies heavily on the authors' own prior work, which is legitimate and not circular in itself: the proposed schemes are evaluated after construction through simulations and weight enumeration. The main correctness risk is the unproven Eq. (8) and the unverified enumeration behind Table I. If the authors can provide a rigorous proof or replace the enumeration with a machine-checked computation, the paper would be in much better shape. As it stands, the central explanatory claim is conditional on an unproven condition."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat'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.\n\nWhere 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.\n\nNone 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.\n\nWho 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.","headline":"Genuinely new code constructions with real but modest gains; the MWC enumeration explaining them is unproven, so the paper deserves review rather than rejection.","tokens_in":10305,"tokens_out":3082,"would_cite":true,"duration_ms":32580,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["polar codes","PAC codes","CRC-polar codes","minimum weight codewords","error coefficient","list decoding","rate profile design","pre-transformation"],"falsifier":"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.","tokens_in":9392,"feed_emoji":"📡","tokens_out":12683,"duration_ms":113526,"temperature":0.7,"pith_summary":"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.","feed_headline":"Freezing the best bit positions buys PAC codes up to 0.5 dB","feed_subtitle":"Parity forced onto the most reliable coordinates breaks the row combos behind minimum-weight codewords.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the theorem that minimum-weight codewords are formed only by the specific row combinations in (5)-(7), the foundation of the coset-by-coset enumeration.","marker":"[17]"},{"why":"States Lemma 1, the limitation of forward convolution (the two incapable-coset conditions) that PS-PAC codes are designed to break.","marker":"[7]"},{"why":"Introduces PAC codes and the convolutional pre-transformation used to turn frozen coordinates into no-freedom bits.","marker":"[3]"},{"why":"Introduces list decoding and CRC-polar codes, the baselines and the decoder used in all simulations.","marker":"[2]"},{"why":"Defines polar codes and the polar transform whose rows carry the entire coset analysis.","marker":"[1]"},{"why":"Shows analytically that convolutional pre-transformation reduces the number of minimum-weight codewords in PAC codes, the effect PS-PAC extends.","marker":"[8]"},{"why":"The authors' earlier reverse PAC codes addressed the second condition of Lemma 1 and motivate attacking the first condition here.","marker":"[16]"},{"why":"Provides the type I / type II error classification used to explain why precoded frozen bits improve list-decoding performance.","marker":"[19]"}],"fun_headline_variants":["Parity on best bits kills min-weight codewords, buys 0.5 dB","Forced CRC parity on reliable coords cuts error coefficient 94%","Why placing parity on reliable bits improves PAC codes by 0.5 dB"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Parity on best bits kills min-weight codewords, buys 0.5 dB","Forced CRC parity on reliable coords cuts error coefficient 94%","Why placing parity on reliable bits improves PAC codes by 0.5 dB"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000382,"raw_usage":{"total_tokens":2030,"prompt_tokens":952,"completion_tokens":1078,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":568,"completion_tokens_details":{"reasoning_tokens":1011}},"tokens_in":568,"tokens_out":1078,"duration_ms":11081,"temperature":1.0,"reasoning_tokens":1011,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T00:43:59.951395+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"On the formation o f min-weight codewords of polar/pac codes and its applications,","cited_arxiv_id":null,"evidence_quote":"Supplies the theorem that minimum-weight codewords are formed only by the specific row combinations in (5)-(7), the foundation of the coset-by-coset enumeration."},{"cited_title":"On the minimum weight codewords o f pac codes: The impact of pre-transformation,","cited_arxiv_id":null,"evidence_quote":"States Lemma 1, the limitation of forward convolution (the two incapable-coset conditions) that PS-PAC codes are designed to break."},{"cited_title":"List decoding of polar codes,","cited_arxiv_id":null,"evidence_quote":"Introduces list decoding and CRC-polar codes, the baselines and the decoder used in all simulations."},{"cited_title":"Channel polarization: A method for construc ting capacity- achieving codes for symmetric binary-input memoryless cha nnels,","cited_arxiv_id":null,"evidence_quote":"Defines polar codes and the polar transform whose rows carry the entire coset analysis."},{"cited_title":"On Convolutional Precoding i n PAC Codes,","cited_arxiv_id":null,"evidence_quote":"Shows analytically that convolutional pre-transformation reduces the number of minimum-weight codewords in PAC codes, the effect PS-PAC extends."},{"cited_title":"Reverse pac codes: Look- ahead list decoding,","cited_arxiv_id":null,"evidence_quote":"The authors' earlier reverse PAC codes addressed the second condition of Lemma 1 and motivate attacking the first condition here."},{"cited_title":"On error probability of crc/polar codes with list decoding,","cited_arxiv_id":null,"evidence_quote":"Provides the type I / type II error classification used to explain why precoded frozen bits improve list-decoding performance."}],"review_version":1}