A Study Of Skew-Polycyclic Codes Over A Non-Chain Ring

Manju Khan; Seema Antil; Seema Chahal; Sugandha Maheshwary

REVIEW 2 major objections 5 minor 30 references

Skew polycyclic codes over a non-chain ring are classified by four generators once the defining polynomial is central.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-07-10 09:43 UTC pith:LAMC2G3S

load-bearing objection Solid incremental classification of skew polycyclic codes over a standard non-chain ring; the four-generator form and free-code criterion are clean under the stated centrality hypotheses, and the examples produce real MDS Gray images. the 2 major comments →

arxiv 2607.08304 v1 pith:LAMC2G3S submitted 2026-07-09 cs.IT math.ITmath.RA

A Study Of Skew-Polycyclic Codes Over A Non-Chain Ring

Seema Antil , Seema Chahal , Manju Khan , Sugandha Maheshwary This is my paper

classification cs.IT math.ITmath.RA MSC 16S3694B0594B1594B60

keywords skew polycyclic codesnon-chain ringsskew constacyclic codesGray mapfinite ringsautomorphismsMDS codes

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper classifies linear codes over the finite non-chain ring of four-term elements with two nilpotents of square zero. It shows that when a monic polynomial is central in the associated skew polynomial ring, every skew polycyclic code of the corresponding length is a left ideal generated by at most four polynomials that satisfy explicit right-divisibility and degree bounds. The same description specialises to skew constacyclic codes of length a power of the characteristic, and further to ordinary constacyclic codes when the automorphism is trivial. Free codes are precisely those generated by a single monic right divisor, and their ranks are immediate. Concrete examples produce Gray images that meet MDS bounds, showing that the algebraic description yields optimal codes over the residue field.

Core claim

Every left ideal of the quotient by a power of a central monic polynomial is uniquely of the form generated by four elements f1+uf12+vf13+uvf14, uf2+vf23+uvf24, vf3+uvf34 and uvf4, where the fi are monic proper divisors (or zero) of the projected power, they form a right-divisibility chain f4 divides each fi which divides f1, and the correction terms have strictly lower degree. Free codes arise exactly when f1 equals f4.

What carries the argument

The unique four-generator form of left ideals in R[x;Theta]/<f^j> under centrality of f (Theorem 3.2), obtained by successive projections onto the u-, v- and uv-torsion and application of the right division algorithm.

Load-bearing premise

The defining polynomial must be central, so that the quotient is two-sided and left ideals coincide with polycyclic codes; without that the whole classification and the later CRT decomposition fail.

What would settle it

Exhibit a non-central monic f for which some left ideal of the quotient cannot be written in the claimed four-generator form, or produce a free left ideal whose generator is not a monic right divisor of f^j.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

Solid incremental classification of skew polycyclic codes over a standard non-chain ring; the four-generator form and free-code criterion are clean under the stated centrality hypotheses, and the examples produce real MDS Gray images.

read the letter

The paper gives a complete generator description for left ideals of the skew polycyclic quotient R[x;Θ]/⟨f^j⟩ when f is central of degree l. Theorem 3.2 is the main new piece: every left ideal is uniquely generated by four elements of the expected shape (f1 + u f12 + v f13 + uv f14, etc.) with the right-divisibility chain f4 |r fi |r f1 and the usual degree bounds on the correction terms. Free codes are precisely those with f1 = f4, recovering the monic-right-divisor criterion and the rank formula jl - k. That is the natural extension of Kewat–Ghosh–Pattanayak (ordinary cyclic over the same ring) and of the chain-ring skew results of Hesari–Samei et al. to the non-chain setting.

The CRT reduction for x^{np^s} - λ under centrality of the irreducible factors is standard but cleanly executed, so the length-p^s classifications for the five unit types become specializations rather than new structural work. The Hensel-lifted factorization when Θ = id and x^n - α0 is reducible likewise extends Cao et al. and Zhao–Tang–Gu from Fpm + u Fpm to the full four-element ring. The Magma examples are honest: several Gray images are MDS of the expected parameters.

Soft spots are minor and already flagged by the authors. Centrality of f (and of x^{np^s} - λ) is indispensable; without it the quotient is not two-sided and the whole correspondence collapses. Several long case lists in Section 4 omit full proofs and simply say “the arguments are the same as …”; that is acceptable for a classification paper but will annoy a referee who wants every generator verified. No circularity, no free parameters, no invented objects.

This is for people who already work on repeated-root codes over finite non-chain rings. It will not change the broader landscape, but it is a self-contained, usable reference for the next person who needs the generator form over this alphabet. I would send it to peer review; a competent referee can check the projections and the uniqueness arguments in a day or two. Worth citing if you are writing about the same ring or about free skew polycyclic codes.

Referee Report

2 major / 5 minor

Summary. The paper studies skew polycyclic codes over the finite non-chain ring R = F_{p^m} + u F_{p^m} + v F_{p^m} + uv F_{p^m} (u^{2} = v^{2} = 0, uv = vu). Under the standing centrality hypothesis on f(x) (Proposition 2.1), it classifies left ideals of the quotient R[x; Θ]/⟨f(x)^j⟩ (Theorem 3.2) via successive projections and torsion submodules, characterizes free codes by monic right divisors (Proposition 3.4 / Corollary 3.5), and reduces skew (λ, Θ)-constacyclic codes of length np^s to these polycyclic components via a CRT decomposition into orthogonal idempotents (Section 3.2). Section 4 specializes the classification to length p^s for the five unit types of λ, while Section 5 treats the commutative case (Θ = id) according to the irreducibility of x^n − α_0, extending earlier results over F_{p^m} + u F_{p^m}. Examples produce Gray images that include MDS codes.

Significance. The work supplies a systematic generator description for skew polycyclic and constacyclic codes over a natural non-chain alphabet that properly contains the chain rings previously treated in the literature. The free-code criterion, rank formula, and CRT reduction are clean and immediately usable for enumeration and construction; the explicit length-p^s lists and Gray-image tables give concrete optimal codes. The extension of the Cao–Zhao results from F_{p^m} + u F_{p^m} to the four-dimensional non-chain ring is a genuine, if incremental, advance for the algebraic theory of repeated-root codes. The centrality hypothesis is stated clearly and is the natural price of working with two-sided ideals in the skew setting.

major comments (2)

Theorem 4.1 (and the parallel Type-II–IV lists for the other unit classes) enumerates dozens of generator configurations without an independent verification that every combination of monic divisors and correction polynomials satisfying the right-divisibility conditions of Theorem 3.2 and Corollary 3.3 actually yields a distinct ideal, or that no further relations collapse some of them. A short uniqueness or cardinality argument (or a reference to a computer check for small p^s) would make the classification load-bearing rather than merely formal.
Section 3.2 assumes that every irreducible factor f_j of the central polynomial x^{np^s} − λ remains central. While Proposition 2.1 gives necessary and sufficient conditions, the paper never verifies that such a complete central factorization exists for a general unit λ and automorphism Θ of order dividing np^s. Without that existence statement the CRT reduction is conditional on an extra hypothesis that is not automatically inherited from the centrality of x^{np^s} − λ.

minor comments (5)

Several arXiv preprints cited as [BMMOa26], [CAMK26], [TS26], [RPM26] carry 2026 dates; if they remain unpublished the bibliographic entries should be updated or flagged as preprints.
In Lemma 3.1 the two determinant conditions α1β2 − α2β1 ≠ 0 and α1β2 + α2β1 ≠ 0 are stated without a short geometric interpretation (invertibility of the linear map on the maximal ideal); a one-line remark would help the reader.
Tables 1–3 list Gray-image parameters but do not record the minimum-distance tables or the Magma commands used; a short reproducibility note would strengthen the experimental claims.
Typographical inconsistencies appear in the notation for the ambient rings (R^{jl}_{u^{2},v^{2},f} versus R^{ps}_{u^{2},v^{2},λ_i}); a uniform convention would improve readability.
The proof of Proposition 5.1 proceeds by induction on degree but never explicitly invokes that the residue field is a field; a parenthetical reminder would make the argument self-contained.

Circularity Check

0 steps flagged

No significant circularity; structure theorems derived from right-division, projections and CRT under explicit centrality hypotheses, with only non-load-bearing self-citations to related preprints.

full rationale

The paper's central claims (Theorem 3.2 generator form for left ideals of R^{jl}_{u^{2},v^{2},f}, free-code characterization in Prop. 3.4/Cor. 3.5, CRT decomposition of skew constacyclic codes, and the five-type classification for length p^s) are obtained by successive application of the right-division algorithm (McDonald), coefficient-wise projections π_v/π_u/μ to known classifications over the chain rings R_{u^{2}} and R_{v^{2}} (cited from HS23), and the Chinese Remainder Theorem for orthogonal idempotents. Centrality of f(x) and of x^{np^s}-λ is stated as an explicit hypothesis (Prop. 2.1) required for the quotient to be two-sided; it is not derived from the conclusions. Free rank formulae follow immediately once the monic right-divisor generator is obtained. Section 4 specializations and the commutative case (Section 5) reuse the same generators plus Hensel's lemma; none of the listed ideal types is forced by a normalization choice or by restating a fitted quantity. Self-citations ([CAMK26] for a weight-preserving map, [BMMOa26] for the definition of skew polycyclic codes) supply auxiliary lemmas or context and are not used as uniqueness theorems that forbid alternatives. The work is therefore self-contained against its stated algebraic assumptions and external benchmarks (MDS parameters obtained via Magma/Gray map).

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The paper is pure algebra; it introduces no free numerical parameters and no new physical or combinatorial entities. All load-bearing ingredients are either standard facts about skew polynomial rings and finite local rings or domain-specific centrality hypotheses needed to make the quotients two-sided. The ledger therefore contains only those background axioms and the centrality assumptions that are repeatedly invoked.

axioms (4)

standard math Right division algorithm holds in R[x;Theta] whenever the leading coefficient of the divisor is a unit (McDonald, Finite Rings with Identity).
Invoked repeatedly to obtain unique remainders of degree less than deg(f_j) and to prove uniqueness of the generators in Theorem 3.2.
domain assumption A monic polynomial f is central in the skew polynomial ring if and only if Theta fixes its coefficients, the coefficients satisfy the twisted commutation relations, and Theta^n = id (Proposition 2.1).
Required for <f^j> to be two-sided so that left ideals correspond to polycyclic codes; assumed throughout Sections 3-5.
domain assumption The units of R_u2,v2,pm fall into exactly five conjugacy classes lambda1=alpha, …, lambda5=alpha+beta u+gamma v+delta uv.
Used to organize the exhaustive classification of length-p^s codes in Section 4.
standard math Hensel's lemma lifts coprime factorizations from the residue field to the local ring R.
Applied in Section 5.2 to obtain the pairwise-coprime factorization of x^{np^s}-lambda over R.

pith-pipeline@v1.1.0-grok45 · 31084 in / 2976 out tokens · 42835 ms · 2026-07-10T09:43:08.715399+00:00 · methodology

0 comments

read the original abstract

For a prime $p$ and a positive integer $m$, let $\mathbb{F}_{p^m}$ be the finite field of cardinality $p^m$, and let $ R_{u^2,v^2,p^m} =\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}+v\mathbb{F}_{p^m} +uv\mathbb{F}_{p^m}, ~ u^2=v^2=0,\ uv=vu, $ be a finite non-chain ring. In this paper, we study skew polycyclic codes of length $lj$ associated with $f(x)^j$, where $f(x)$ is a central polynomial of degree $l$ in $R_{u^2, v^2, p^m}[x; \Theta],$ where $\Theta$ being an automorphism of $R_{u^2,v^2,p^m}$. We describe these codes, characterize free skew polycyclic codes, and determine their ranks. Under suitable centrality assumptions, we decompose the quotient ring associated with $x^{np^s}-\lambda$, where $\gcd(n,p)=1$ and $\Theta(\lambda)=\lambda$. This reduces the study of skew $(\lambda,\Theta)$-constacyclic codes of length $np^s$ to the study of left ideals of $\frac{R_{u^2,v^2,p^m}[x;\Theta]}{\langle f(x)^j\rangle}, $ where $f(x)$ is a central irreducible divisor of degree $l$ of $x^{np^s}-\lambda$, for an invertible element $\lambda\in R_{u^2,v^2,p^m}$ and $j\in\mathbb{N}$. We then apply these results to skew $(\lambda,\Theta)$-constacyclic codes of length $p^s$ for different classes of units $\lambda$. Several examples are presented to illustrate the theory and to obtain optimal codes. Finally, when $\Theta$ is the identity automorphism, we study constacyclic codes of length $np^s$ over $R_{u^2,v^2,p^m}$, according as $x^n-\alpha_0$ is irreducible or reducible over $\mathbb{F}_{p^m}$. These results extend the work of \cite{CCDF18} and \cite{ZTG18} on constacyclic codes of length $np^s$ over $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$ to the finite non-chain ring $R_{u^2,v^2,p^m}$.

discussion (0)

Reference graph

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Chahal, S. and Maheshwary, S. , TITLE =. Finite Fields Appl. , FJOURNAL =. 2025 , PAGES =

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, TITLE =

Dinh, H.Q. , TITLE =. Discrete Math. , FJOURNAL =. 2013 , NUMBER =

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Dinh, H.Q. , TITLE =. Ring theory and its applications , SERIES =. 2014 , ISBN =

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Skew polycyclic over finite chain rings associated to trinomials , author=. 2026 , eprint=

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Hesari, R.M. and Mohebbei, M. and Rezaei, R. and Samei, K. , TITLE =. Commun. Korean Math. Soc. , FJOURNAL =. 2024 , NUMBER =

work page 2024

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and Ling, S

Jitman, S. and Ling, S. and Udomkavanich, P. , TITLE =. Adv. Math. Commun. , FJOURNAL =. 2012 , NUMBER =

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, TITLE =

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and Raj, R

Pathak, S. and Raj, R. and Maity, D. , TITLE =. Cryptogr. Commun. , FJOURNAL =. 2025 , NUMBER =

work page 2025

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and Tang, X

Zhao, W. and Tang, X. and Gu, Z. , TITLE =. Finite Fields Appl. , FJOURNAL =. 2018 , PAGES =

work page 2018

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and Hesari, R.M

Bagheri, S. and Hesari, R.M. and Rezaei, H. and Samei, K. , TITLE =. Iran. J. Sci. Technol. Trans. A Sci. , FJOURNAL =. 2022 , NUMBER =

work page 2022

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and Samei, K

Hesari, R.M. and Samei, K. , TITLE =. Iran. J. Sci. , FJOURNAL =. 2026 , NUMBER =

work page 2026

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and Cao, Y

Cao, Y. and Cao, Y. and Dinh, H.Q. and Fu, F.W. and Gao, Jian and Sriboonchitta, Songsak , TITLE =. Adv. Math. Commun. , FJOURNAL =. 2018 , NUMBER =

work page 2018

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and Cannon, J

Bosma, W. and Cannon, J. and Playoust, C. , TITLE=. Journal of Symbolic Computation , VOLUME=. 1997 , NUMBER=

work page 1997

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Kewat, P. K. and Ghosh, B. and Pattanayak, S. , TITLE =. Finite Fields Appl. , FJOURNAL =

work page

[23] [23]

and Pathak, S

Raj, R. and Pathak, S. and Maity, D. , TITLE =. Comput. Appl. Math. , FJOURNAL =. 2026 , NUMBER =

work page 2026

[24] [24]

and Kewat, P.K

Dinh, H.Q. and Kewat, P.K. and Kushwaha, S. and Yamaka, W. , TITLE =. Discrete Math. , FJOURNAL =. 2020 , NUMBER =

work page 2020

[25] [25]

Skew Constacyclic Codes Of Length $np^s$ over $ \frac{\mathbb{F}_{p^m}[u]}{\langle u^k \rangle}

Chahal, S. and Antil, S. and Maheshwary, S. and Khan, M. , year=. Skew Constacyclic Codes Of Length. 2605.15925 , archivePrefix=

work page internal anchor Pith review Pith/arXiv arXiv

[26] [26]

Skew Polycyclic Codes over $\frac{\mathbb{F}_{p^m}[u]}{\langle u^t \rangle}$

Tiwari, A. and Sarma, R. , year=. Skew Polycyclic Codes Over. 2605.13020 , archivePrefix=

work page internal anchor Pith review Pith/arXiv arXiv

[27] [27]

and Niu, C

Si, X. and Niu, C. , TITLE =. AIMS Math. , FJOURNAL =. 2023 , NUMBER =

work page 2023

[28] [28]

and Karadeniz, S

Nuh, A. and Karadeniz, S. and Yildiz, B. , TITLE =. Appl. Algebra Engrg. Comm. Comput. , FJOURNAL =. 2013 , NUMBER =

work page 2013

[29] [29]

and Karadeniz, S

Yildiz, B. and Karadeniz, S. , TITLE =. Des. Codes Cryptogr. , FJOURNAL =. 2011 , NUMBER =

work page 2011

[30] [30]

and Karadeniz, S

Dougherty, S.T. and Karadeniz, S. and Yildiz, B. , TITLE =. Des. Codes Cryptogr. , FJOURNAL =. 2012 , NUMBER =

work page 2012