{"id":"31988d02-ef6d-4c3a-9085-6ea8c5a9628b","arxiv_id":"2506.15467","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"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.","lead":"Polar codes are usually chosen to maximize channel reliability, which leaves their set of very low-weight codewords poor and hurts error rates. The authors propose a rule-based adjustment to the standard 5G reliability ordering, guided by each position's contribution to low-weight codewords, and report gains for several polar and PAC code rates.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Algorithm 2 as printed removes the lowest-|λf| degree-r rows, not the highest as the text claims; the stated weight-distribution mechanism and the 5G-WD results are not reproducible from the pseudocode.","rationale":"The central promise is a search-free rule that improves weight distribution by swapping specifically chosen degree-r rows. For that promise to be true, the algorithm printed in the paper must actually remove the rows the theory says are harmful. It does not: FlatenZs sorts by |λf| ascending, and line 8 takes the head of that list for deletion. This is not a subtle numerical issue; it is a deterministic contradiction with the prose. Even if the intended variant was used to generate Figs. 3–4, the paper as submitted does not let a reader reproduce the claimed construction. The decreasing-monomial issue is related but secondary: if the swapped set is not decreasing, the formulas (1)–(2) that motivate the choice of rows no longer apply; however the empirical BLER comparison could in principle survive. The algorithm-order inconsistency, by contrast, strikes at the definition of the construction itself. I therefore recommend keeping the CONDITIONAL verdict, but the condition should be a corrected Algorithm 2 and a verification that the reported 5G-WD profiles correspond to the stated selection rule.","tokens_in":9767,"tokens_out":16035,"duration_ms":170142,"concrete_test":"Implement Algorithm 2 verbatim for the N and K values of Figs. 3–4 using the 3GPP 5G ordering, and print the |λf| values of Qwmin[1:π] and Qwmin[|Qwmin|-π+1:|Qwmin|]. If the former are the five smallest and the latter the five largest, the pseudocode contradicts the prose. Then re-run the BLER simulations with the intended maximum-|λf| removal; if the figure curves require the intended variant, the paper must be revised before the claimed 5G-WD construction is accepted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section V, Algorithm 2 is internally inconsistent with its surrounding text. FlatenZs (lines 10–13) concatenates levels d=0,...,s(m−s), i.e., in increasing |λf|, and within each level sorts by descending reliability. Line 8 then sets I ← (QK \\ Qwmin[1:π]) ∪ Q2wmin[1:π], so the removed degree-r rows are the first π elements of Qwmin: the rows with the smallest |λf| and, within that low-contribution level, the highest reliability. The prose says the algorithm removes 'π elements with degree r and maximum |λf| and minimum reliability.' Thus the construction as written keeps the high-|λf| rows that dominate the minimum-weight sum in Eq. (1) and discards the low-|λf| rows, which is the opposite of the stated weight-distribution rationale. Because all reported 5G-WD profiles are defined by this pseudocode, either the figures were produced by a different selection rule or the algorithm in the paper is not the one that was simulated. A second, related gap is that nothing in the swap checks that the new information set is still decreasing: an inserted degree-(r−1) monomial can have a missing divisor, in which case the polar weight-enumeration formulas (1)–(2) used to justify the swap do not apply. Both issues need to be fixed before the claimed improvement can be attributed to a 'weight-contribution partial order' rule.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies polar codes as decreasing monomial codes and proposes to modify the information set obtained from the 5G reliability sequence by swapping a small number of degree-r monomials with degree-(r-1) monomials chosen according to a weight-contribution order based on the parameter |λ_f|. The proposed construction (Algorithm 2) requires no search. Numerical results are presented for N=128 and N=256 at rates 1/4, 1/2, and 3/4 under CRC-aided SCL and PAC/SCL decoding, together with minimum-distance and minimum-weight-codeword counts in Table I. The central claim is that the resulting '5G-WD' rate profiles improve BLER over the unmodified 5G profile, particularly for PAC codes, while also improving d_min in several cases.","tokens_in":10038,"tokens_out":9324,"duration_ms":89197,"significance":"If the construction rule works as claimed, the paper provides a simple, search-free way to improve the weight distribution of polar-like codes, which would be a useful practical contribution to polar/PAC code design. The paper's algebraic framework—partitioning monomials by degree and by |λ_f| level, with the symmetry relation |λ_f|+|λ_\\check f|=s(m-s)—is plausible and builds on prior work in a coherent way. However, the significance is currently tempered by two load-bearing gaps: the printed algorithm appears to select the opposite set from the one described in the text, and the validity of the weight-enumeration formulas for the modified information sets is not established. The empirical gain, if reproducible after the algorithm is corrected, would justify publication; as it stands, the central mechanism is not reliably connected to the reported results.","major_comments":[{"comment":"The pseudocode contradicts the surrounding prose in a load-bearing way. FlatenZs (lines 10–13) concatenates the contribution levels in increasing d, i.e., in increasing |λ_f|, and sorts within each level by descending reliability. Therefore Qwmin[1:π] in line 8 consists of the degree-r rows with the smallest |λ_f|, not the rows with maximum |λ_f| as the text claims. Because Eq. (1) shows |W_wmin| is controlled by the sum of 2^{r+|λ_f(f)|} over I_r, removing the low-contribution rows leaves the dominant high-contribution rows in place; the stated mechanism does not match the code that is supposedly simulated. The authors must correct this inconsistency and state which selection rule actually generated the 5G-WD curves in Figs. 3–4.","section":"Section V, Algorithm 2"},{"comment":"The swap in line 8 is not checked for decreasingness. The weight-enumeration formulas (1)–(2) and Remark 2 apply to decreasing monomial codes, but inserting a degree-(r−1) monomial whose divisors are not all present in I produces a non-decreasing set. No proof or verification is provided that the 5G-WD information sets remain decreasing for the reported parameters. Without this property, the formulas on which the design rationale rests no longer apply, so the authors need to prove closure, enforce it in the algorithm, or at least verify it numerically for every reported (N,K) pair.","section":"Section II-B and Section V, Algorithm 2"},{"comment":"The central empirical claim is not sufficiently documented for reproduction. The reported simulations omit the PAC pre-transformation polynomial, the CRC length and generator used for CRC-Polar and for the PAC CRC, the exact π values chosen for each rate (π_max=5 is only an upper bound), the channel model, and the number of simulated frames or error bars. In addition, the abstract's statement that improvements hold 'both with and without pre-transformation' is not supported by the figures, which compare only CRC-Polar and PAC schemes; there is no untransformed polar-code 5G versus 5G-WD curve. Finally, Table I shows that for N=256, K=192 the 5G-WD profile has the same d_min and Admin as the unmodified PAC profile, so the source of the reported gain in that case needs explanation.","section":"Section V, Figs. 3–4 and Table I"}],"minor_comments":[{"comment":"The expression min(Qwmin, Q2wmin) is undefined as written; it should be min(|Qwmin|, |Q2wmin|), and the indexing [1:π] should specify whether it is 1-based.","section":"Algorithm 2, line 7"},{"comment":"The relation f ≤_{wmin} g defined by |λ_f| ≤ |λ_g| is not a partial order because it lacks antisymmetry; 'preorder' or 'total preorder with equivalence classes' would be more accurate.","section":"Definition 4"},{"comment":"The manuscript contains numerous typographical and spacing errors (for example in the Abstract and Section I) and several sentences are grammatically incomplete; a careful language edit is needed.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The paper's main novelty is incremental and heavily self-referential, but the underlying idea of contribution-level-based selection is reasonable. The decisive issue is whether the reported simulations correspond to a corrected version of Algorithm 2; if the pseudocode is simply fixed without re-running the experiments, the empirical claims cannot be trusted. I would encourage the editor to require a consistency check between the described algorithm, the pseudocode, and the simulation code, and to request the missing experimental details before any further decision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here is my read on arXiv:2506.15467. The paper has a genuinely attractive idea: instead of searching over polar code constructions, swap a few degree-r rows of the 5G reliability sequence for degree-(r-1) rows selected by the authors' |λf| weight-contribution measure. The math behind that measure — especially the symmetry of |λf| with the complement monomial and the level structure of degree-s monomials — is clearly explained and is a legitimate minor contribution.\n\nThe problem is the algorithm. In Section V, the prose says line 8 removes the π degree-r elements with maximum |λf| and minimum reliability, replacing them with degree-(r-1) elements with minimum |λf|. But FlatenZs flattens each degree's levels in increasing |λf|, so Qwmin[1:π] are the elements with the smallest |λf|. Line 8 therefore removes the low-contribution rows and keeps the high-contribution ones — the exact opposite of the stated weight-distribution rationale. Either the figures were produced by a different selection rule, or the pseudocode is not the algorithm that was run. As printed, the 5G-WD profiles are not reproducible.\n\nThere is a second, independent gap. Nothing in the swap verifies that the new information set is still decreasing. The added degree-(r-1) monomials come from outside QK, so their divisors may not be present. If a divisor is missing, the weight-enumeration formulas (1)-(2) no longer apply, and the whole justification collapses. The authors need to either prove that every inserted monomial's divisors are already in I, or restrict the swap to monomials that satisfy this.\n\nMinor points: no simulation details, no error bars, no code; the gains are shown only for N=128 and 256 at three rates. That is a thin base, but not fatal for a short paper.\n\nIf the authors fix the algorithm description and add the decreasing-monomial check, this could be a useful practical note for the polar code community. As it stands, the central empirical claim is not trustworthy. I would send it to review because the underlying idea is not evidently wrong and the fix seems plausible, but I would make it clear that the current version needs substantial revision before the results can be taken at face value.","headline":"Search-free polar-code tweak has a plausible idea, but Algorithm 2 as printed contradicts its own prose and the decreasing-monomial check is missing.","tokens_in":10589,"tokens_out":7416,"would_cite":false,"duration_ms":66403,"reading_group":"maybe","serious_thinker":"unclear","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94B05","94B35"],"pacs":[],"model":"deepseek-v4-flash","headline":"Replacing a handful of high-degree monomials in the 5G polar information set with lower-degree, low-contribution monomials improves the code's weight distribution and error rate without search.","keywords":["polar codes","decreasing monomial codes","Reed-Muller codes","weight distribution","code construction","PAC codes","SCL decoding","minimum distance"],"falsifier":"Run Algorithm 2 on a concrete case such as $N=256$, $K=192$, and check each inserted degree-$(r-1)$ monomial for divisor closure against the resulting information set. If some divisor is missing, exhaustively enumerate the code's weight spectrum; if the true $d_{\\min}$ or $A_{d_{\\min}}$ disagrees with the decreasing-monomial formula, the paper's justification for the swap fails for that case, and the simulated BLER gain would need to stand on other grounds.","tokens_in":9543,"feed_emoji":"🔁","tokens_out":10074,"duration_ms":98130,"temperature":0.7,"pith_summary":"Polar codes are normally built by picking the most reliable sub-channels, but that reliability ranking pays no attention to the code's weight distribution. This paper argues the two can be balanced by a tiny deterministic edit: remove up to five building-block monomials from the top degree of the 5G information set that contribute most to low-weight codewords, and replace them with the same number of one-degree-lower monomials that contribute least. The edited profile has better minimum distance and fewer low-weight codewords, and under PAC/SCL decoding it shows lower block error rates at lengths 128 and 256 across rates $1/4$, $1/2$, $3/4$. The point of the construction is that it uses no search or optimization, only a precomputed weight-contribution ordering together with the 5G reliability sequence.","feed_headline":"Swapping five monomials improves polar codes with no search","feed_subtitle":"A deterministic tweak to the 5G reliability sequence cuts BLER and lifts minimum distance under PAC/SCL decoding","key_machinery":"The central object is the weight-contribution parameter $|\\lambda_f|$ for a monomial $f$ of degree $s$: for each variable in $f$, count how many lower-index variables are absent from $f$, and sum those counts. Smaller $|\\lambda_f|$ means fewer minimum-weight codewords contributed by $f$, per the enumeration formulas (1)-(2); the paper also proves the symmetry $|\\lambda_f|+|\\lambda_{\\check f}|=s(m-s)$. This parameter organizes monomials of each degree into levels in Algorithm 1, and Algorithm 2 flattens those levels, interleaving reliability order with weight-contribution order to decide which monomials to remove and which to insert. The decreasing-monomial structure of polar codes is what lets these formulas and the SC error expression apply.","core_discovery":"On the paper's own terms, the discovery is that the weight contribution of a monomial, measured by the parameter $|\\lambda_f|$, supplies a usable construction order alongside reliability. Algorithm 2 removes $\\pi_{\\max}=5$ monomials of degree $r$ from the 5G information set that have the largest $|\\lambda_f|$ and the weakest reliability within their level, and inserts the same number of degree-$(r-1)$ monomials with the smallest $|\\lambda_f|$ taken from outside the information set. The resulting '5G-WD' profile improves the minimum distance and the low-weight spectrum in the tested cases and yields lower BLER than the unmodified 5G profile under PAC/SCL decoding, both with and without pre-transformation. Because the choice is read off from a precomputed contribution structure $Z_s$ and the 5G ordering, no search or optimization is involved.","pith_inferences":["A testable extension the paper leaves open is to prove that Algorithm 2's swap preserves the decreasing property for every $m$ and $\\pi_{\\max}$, not just the simulated instances; one could check all inserted monomials for divisor closure and, where it fails, quantify how much of the gain disappears.","The symmetry $|\\lambda_f|+|\\lambda_{\\check f}|=s(m-s)$ hints at a dual construction for high-rate codes: swapping in the opposite direction, from low-degree monomials with large $|\\lambda_f|$ to higher-degree ones, might balance the weight spectrum near rate one, a regime the paper does not test.","The choice $\\pi_{\\max}=5$ is empirical; sweeping $\\pi$ at fixed $N$ and $K$ would reveal where the reliability loss from replacing degree-$r$ sub-channels overtakes the weight-spectrum gain, and whether the optimum depends on $N$ and rate."],"forward_implications":["The same deterministic swap can be applied to any 5G polar rate profile at a given $N$ and $K$ without retraining or search, with reported gains across rates $1/4$, $1/2$, $3/4$ and lengths $128$ and $256$.","The weight-enumeration formulas cease to be only analytic tools and become design criteria: minimizing $|\\lambda_f|$ at the top degree is a proxy for minimizing $A_{d_{\\min}}$ and $A_{1.5 d_{\\min}}$.","Because the swap keeps the dimension $K$ unchanged and only moves the boundary between degrees $r$ and $r-1$, it can be combined with CRC or PAC pre-transformations, and the paper shows PAC with the modified profile outperforming PAC with the standard 5G profile.","The reported $d_{\\min}$ and $A_{d_{\\min}}$ values indicate the trade-off is not a strict improvement in both quantities: some CRC-polar profiles keep smaller $A_{d_{\\min}}$ but lose on reliability, which supports the paper's emphasis on balancing reliability with weight distribution."],"supporting_citations":[{"why":"Establishes polar codes as decreasing monomial codes, the algebraic setting the design uses.","marker":"[5]"},{"why":"Supplies the closed-form count of $1.5 d_{\\min}$ codewords used to motivate reducing $|\\lambda_f|$.","marker":"[6]"},{"why":"Extends weight enumeration to weights below $2 d_{\\min}$, supporting the focus on the low-weight spectrum.","marker":"[7]"},{"why":"Introduces the weight-contribution partial order and the $|\\lambda_f|$ levels that Algorithm 1 builds on.","marker":"[8]"},{"why":"Defines the polarization-weight reliability metric used for ordering within levels.","marker":"[25]"},{"why":"Gives the SC decoding error expression used to argue that reliability must be balanced with weight distribution.","marker":"[20]"},{"why":"Introduces polar codes and the sub-channel reliability construction that the paper modifies.","marker":"[1]"}],"fun_headline_variants":["Swap 5 monomials, lift polar code min distance and BLER","Deterministic monomial swap fixes polar code weight spectrum","Swap 5 monomials: polar code min distance up, BLER down","No-search tweak to 5G polar codes improves weight distribution","Lift polar code min distance with deterministic 5-monomial swap"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction assumes that after swapping monomials the information set is still closed under divisors; if a divisor of an inserted monomial is missing, the code is no longer a polar-like decreasing monomial code and the weight formulas used to justify the swap no longer apply.","fun_headline_variants_meta":{"raw":{"variants":["Swap 5 monomials, lift polar code min distance and BLER","Deterministic monomial swap fixes polar code weight spectrum","Swap 5 monomials: polar code min distance up, BLER down","No-search tweak to 5G polar codes improves weight distribution","Lift polar code min distance with deterministic 5-monomial swap"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001075,"raw_usage":{"total_tokens":4442,"prompt_tokens":827,"completion_tokens":3615,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":443,"completion_tokens_details":{"reasoning_tokens":3520}},"tokens_in":443,"tokens_out":3615,"duration_ms":25028,"temperature":1.0,"reasoning_tokens":3520,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T19:34:31.751020+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run Algorithm 2 on a concrete case such as $N=256$, $K=192$, and check each inserted degree-$(r-1)$ monomial for divisor closure against the resulting information set. If some divisor is missing, exhaustively enumerate the code's weight spectrum; if the true $d_{\\min}$ or $A_{d_{\\min}}$ disagrees with the decreasing-monomial formula, the paper's justification for the swap fails for that case, and the simulated BLER gain would need to stand on other grounds.","supporting_citations":[{"cited_title":"Alge braic properties of polar codes from a new polynomial formalism,","cited_arxiv_id":null,"evidence_quote":"Establishes polar codes as decreasing monomial codes, the algebraic setting the design uses."},{"cited_title":"On the distribution of weights less than 2w min in polar codes,","cited_arxiv_id":null,"evidence_quote":"Extends weight enumeration to weights below $2 d_{\\min}$, supporting the focus on the low-weight spectrum."},{"cited_title":"Weight structur e of low/high-rate polar codes and its applications,","cited_arxiv_id":null,"evidence_quote":"Introduces the weight-contribution partial order and the $|\\lambda_f|$ levels that Algorithm 1 builds on."},{"cited_title":"Beta-expansion: A theoretical framework fo r fast and recursive construction of polar codes,","cited_arxiv_id":null,"evidence_quote":"Defines the polarization-weight reliability metric used for ordering within levels."},{"cited_title":"Performance and construction of polar codes on symmetric binary-input memoryless channels,","cited_arxiv_id":null,"evidence_quote":"Gives the SC decoding error expression used to argue that reliability must be balanced with weight distribution."}],"review_version":2}