Towards Weight Distribution-Aware Polar Codes

Mohammad Rowshan; Vlad-Florin Dragoi

REVIEW 3 major objections 3 minor 29 references

Towards Weight Distribution-Aware Polar Codes

T0 review · 3 major / 3 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2506.15467 v1 pith:444VGC2H submitted 2025-06-18 cs.IT math.IT

Mohammad Rowshan , Vlad-Florin Dragoi This is my paper · ORCID

classification cs.ITmath.IT MSC 94B0594B35

keywords polarcodesdecreasingmonomialReed-MullerweightdistributioncodeconstructionPACSCLdecodingminimumdistance

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

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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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

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.

The 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.

There 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.

Minor 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.

If 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.

Referee Report

3 major / 3 minor

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.

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 (3)

[Section V, Algorithm 2] 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 II-B and Section V, Algorithm 2] 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 V, Figs. 3–4 and Table I] 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.

minor comments (3)

[Algorithm 2, line 7] 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.
[Definition 4] 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.
[General] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the 5G-WD construction is benchmarked independently and the weight-distribution rationale is not just the output re-labeled as input.

full rationale

The claimed derivation is self-contained in the relevant sense: the 5G reliability order Q defines the initial information set; Equations (1)-(3) quantify how |λf| contributes to small-weight multiplicities; Algorithm 2 proposes swapping a bounded number (πmax=5) of degree-r positions with degree-(r-1) positions; and the claimed improvement is then checked by independent minimum-distance computation and by SCL/PAC decoding against external 5G and CRC-polar baselines. No parameter of the algorithm is fitted to the reported BLER curves, and the improvement does not follow by definition from the selection rule alone, since the BLER and dmin values are separate outputs. The self-citations to the authors' earlier weight-enumeration results are parameter-free published results, not an unverified uniqueness theorem used to forbid alternatives, and the paper redefines the λf order and its levels in Definitions 3-4 and Lemma 1, so those citations are independent support rather than a circular premise. The notable issue in this paper is an internal inconsistency: the prose says Algorithm 2 removes degree-r rows with maximum |λf| while the pseudocode, which flattens levels from d=0 upward and takes Qwmin[1:π], removes the minimum-|λf| rows; this is a reproducibility and correctness risk, not a circular dependence.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim rests on two borrowed ingredients: the weight-contribution measure λf from the authors' earlier work, and the polarization-weight reliability metric from [25]. The only hand-chosen numerical parameter is π_max=5. No new physical or mathematical entities are introduced. The main unstated assumption is that the swapped information set remains a decreasing monomial code so the polar weight-enumeration analysis still applies.

free parameters (2)

π_max = 5
Maximum number of positions swapped by Algorithm 2, chosen by the authors for the numerical section. No sensitivity analysis is provided, and all reported BLER gains depend on this cap.
β (beta-expansion exponent) = 2^(1/4)
Exponent in the polarization-weight reliability metric PW(i) adopted from [25]; the ordering produced by Algorithm 2 depends on this choice, and the paper itself notes the metric 'lacks accuracy'.

assumptions (4)

domain assumption The closed-form weight formulas (1) and (2) for |W_{wmin}| and |W_{1.5wmin}|, taken from [5]-[8], are valid for the information sets under consideration.
Invoked in Remark 2 and Section V to justify that lowering |λf| improves the weight distribution. The paper does not re-derive them and their validity requires the code to remain a decreasing monomial code.
domain assumption The polarization weight PW(i) with β=2^(1/4) is an adequate reliability ordering for deciding which sub-channels are weak and which replacement candidates are best.
Section III adopts PW for tractability and explicitly states it 'lacks accuracy'; the swap algorithm's reliability ranking of replacement candidates depends on this choice.
domain assumption The union-Bhattacharyya bound (4), truncated to small weights, predicts practical SCL decoding performance.
Used in Section II-B to motivate the focus on minimum-weight codewords and to justify that reducing |W_{wmin}| improves BLER.
domain assumption The initial 5G reliability sequence Q, when intersected with monomials, defines a decreasing monomial set and remains compatible with the monomial partial order after modification.
Needed for formulas (1)-(2) and polar SC properties; the paper does not verify that Algorithm 2's swapped set is still decreasing.

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Cite this review

Pith. "Pith review of Towards Weight Distribution-Aware Polar Codes." pith.science (2026). https://pith.science/paper/444VGC2H

@misc{pith2026250615467,
  author       = {Pith},
  title        = {Pith review of: Towards Weight Distribution-Aware Polar Codes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/444VGC2H}},
  note         = {Machine review of arXiv:2506.15467}
}

read the original abstract

Polar codes are constructed based on the reliability of sub-channels resulting from the polarization effect. However, this information-theoretic construction approach leads to a poor weight distribution. To address this issue, pre-transformed polar codes, such as CRC-polar codes and PAC codes, have been employed. In this paper, we focus on the structure of polar codes without applying any pre-transformations and explore methods, guided by the weight-contribution partial order, to design polar-like codes with enhanced weight distribution, notably without employing any search or optimization algorithms. Numerical results demonstrate improvement over a range of codes both with and without pre-transformation.

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Works this paper leans on

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Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

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

E. Arıkan, “Channel polarization: A method for construc ting capacity- achieving codes for symmetric binary-input memoryless cha nnels,” IEEE Trans. Inf. Theory , vol. 55, no. 7, pp. 3051–3073, Jul. 2009

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[2] [2]

Crypt- analysis of the McEliece public key cryptosystem based on po lar codes,

M. Bardet, J. Chaulet, V . Dragoi, A. Otmani, and J.-P . Til lich, “Crypt- analysis of the McEliece public key cryptosystem based on po lar codes,” in Post-Quantum Cryptography, vol. 9606, 2016, pp. 118–143

work page 2016

[3] [3]

An algebraic approach for the resolution o f algorithmic problems raised by cryptography and coding theory,

V . F. Dragoi, “An algebraic approach for the resolution o f algorithmic problems raised by cryptography and coding theory,” Ph.D. d isserta- tion, Normandie Universit´ e, 2017

work page 2017

[4] [4]

On the weight structure of Reed- Muller codes,

T. Kasami and N. Tokura, “On the weight structure of Reed- Muller codes,” Trans. Inf. Theory , vol. 16, no. 6, pp. 752–759, Nov. 1970

work page 1970

[5] [5]

Alge braic properties of polar codes from a new polynomial formalism,

M. Bardet, V . Dragoi, A. Otmani, and J.-P . Tillich, “Alge braic properties of polar codes from a new polynomial formalism,” in 2016 IEEE Int. Symp. Inf. Theory (ISIT) , 2016, pp. 230–234

work page 2016

[6] [6]

On the closed-fo rm weight enumeration of polar codes: 1.5d -weight codewords,

V .-F. Dr˘ agoi, M. Rowshan, and J. Y uan, “On the closed-fo rm weight enumeration of polar codes: 1.5d -weight codewords,” IEEE Trans- actions on Communications , vol. 72, no. 10, pp. 5972–5987, 2024. DOI : 10.1109/TCOMM.2024.3394749

arXiv 2024

[7] [7]

On the distribution of weights less than 2w min in polar codes,

Z. Y e, Y . Li, H. Zhang, J. Wang, G. Y an, and Z. Ma, “On the distribution of weights less than 2w min in polar codes,” IEEE Transactions on Communications , 2024

work page 2024

[8] [8]

Weight structur e of low/high-rate polar codes and its applications,

M. Rowshan, V .-F. Dr˘ agoi, and J. Y uan, “Weight structur e of low/high-rate polar codes and its applications,” in 2024 IEEE Int. Symp. on Inf. Theory (ISIT) , 2024, pp. 2945–2950

work page 2024