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

Manju Khan; Seema Antil; Seema Chahal; Sugandha Maheshwary

arxiv: 2605.15925 · v1 · pith:7ELPX5GPnew · submitted 2026-05-15 · 💻 cs.IT · math.IT· math.RA

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

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

Pith reviewed 2026-05-19 19:33 UTC · model grok-4.3

classification 💻 cs.IT math.ITmath.RA

keywords skew constacyclic codesskew polynomial ringsfinite chain ringspolycyclic codesleft idealsconstacyclic codesoptimal parameters

0 comments

The pith

Skew constacyclic codes of length np^s over the ring R_k reduce to skew polycyclic codes of length jl associated with a central irreducible divisor f(x)^j of x^{np^s} - λ.

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

The paper develops a unified method for examining skew constacyclic codes of length np^s over the finite chain ring R_k = F_{p^m}[u]/(u^k) by shifting attention to skew polycyclic codes tied to powers of central irreducible factors. When the multiplier λ lies in the base field, the left ideals of the quotient ring R_k[x; Θ]/ are fully classified for suitable automorphisms Θ, and an isomorphism is established between certain skew cyclic and skew constacyclic families. Concrete analyses are supplied for lengths 3p^s and 6p^s, including explicit factorizations and examples of codes attaining optimal parameters. A sympathetic reader would value this because the reduction replaces a direct search over long code lengths with the simpler algebraic structure of a smaller quotient ring, potentially making enumeration and construction feasible for larger s.

Core claim

Skew constacyclic codes of length np^s over R_k reduce to the study of skew polycyclic codes of length jl associated with a central irreducible divisor f(x)^j of x^{np^s} - λ. For λ in F_{p^m} the left ideals of the quotient R_k[x; Θ]/<f(x)^j> can be classified completely under suitable conditions on Θ, yielding an isomorphism between skew cyclic and skew constacyclic codes; the same framework supplies complete case-by-case factorizations and ideal descriptions for the special lengths 3p^s and 6p^s.

What carries the argument

The central irreducible divisor f(x) of degree l and multiplicity j in the skew polynomial ring R_k[x; Θ], whose power f(x)^j generates the quotient ring whose left ideals correspond exactly to the skew constacyclic codes.

If this is right

All left ideals of R_k[x; Θ]/<f(x)^j> are enumerated when λ lies in the base field, giving an explicit description of every skew constacyclic code.
An explicit ring isomorphism maps skew cyclic codes onto skew constacyclic codes under the stated conditions on Θ.
Complete factorizations of x^{3p^s} - λ, x^{6p^s} - 1 and x^{6p^s} + 1 yield all ideals for those lengths.
Families of codes attaining optimal parameters are constructed via the ideal correspondence.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The reduction may extend to other chain rings or to non-central divisors, allowing similar classifications without requiring centrality.
The isomorphism between cyclic and constacyclic families suggests that distance bounds or decoding algorithms developed for one family transfer directly to the other.
For large s the smaller quotient ring of degree jl offers a practical route to exhaustive search that would be intractable on the original length np^s.

Load-bearing premise

A central irreducible divisor f(x) of degree l and multiplicity j must exist in R_k[x; Θ] such that the left ideal lattice of the quotient by f(x)^j is identical to the lattice of skew constacyclic codes.

What would settle it

For concrete p, m, k, n, s and a chosen automorphism Θ, compute the left ideals in R_k[x; Θ]/<f(x)^j> and check whether they fail to match the set of all skew constacyclic codes of length np^s over R_k; or verify that no such central irreducible f(x) divides x^{np^s} - λ for some invertible λ.

read the original abstract

Let $\mathbb{F}_{p^m}$ be the field containing $p^m$ elements where $p$ is an odd prime and $m \in \mathbb{N}$. In this article, we propose a unified approach to the study of skew constacyclic codes of length $np^s$ over the ring $R_k = \mathbb{F}_{p^m}[u]/\langle u^k \rangle,$ where $n, s, k \in \mathbb{N}$ and $\gcd(n, p)=1$. Consider the skew polynomial ring $R_k[x;\Theta]$, where $\Theta$ is an automorphism of $R_k$ such that $xa = \Theta(a)x$ for all $a \in R_k$. Let $f(x)$ be a central irreducible divisor of $x^{np^s} - \lambda$ of degree $l$ and multiplicity $j$ in $R_k[x;\Theta]$, where $\lambda $ is an invertible element in $R_k$. In this article, we study skew constacyclic codes of length $np^s$ over $R_k$, which reduces to the study of skew polycyclic codes of length $jl$ associated with a polynomial $f(x)^j$. Using the fact that skew polycyclic codes associated with a polynomial $f(x)^j$ can be described by the left ideal structure of the quotient ring $R_k[x;\Theta]/\langle f(x)^{j}\rangle$, we investigate this class of codes for specific choices of $\Theta$. In particular, if $\lambda$ is an invertible element of $\mathbb{F}_{p^m}$, we classify all left ideals and establish an isomorphism between skew cyclic and skew constacyclic codes, under suitable conditions. Furthermore, we provide a comprehensive analysis of skew constacyclic codes of length $3p^s$ over $R_k$. Finally, we examine skew cyclic and skew negacyclic codes of length $6p^s$ over $R_k$ using the factorization of $x^{6p^s} - 1$ and $x^{6p^s} + 1$, respectively; with a complete case-by-case analysis. Examples demonstrating codes with optimal parameters are also included.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper reduces skew constacyclic codes of length np^s over chain rings to polycyclic ones via a central irreducible factor and gives explicit classifications for lengths 3p^s and 6p^s, but the reduction's scope depends on an assumption that needs checking.

read the letter

The main thing here is a reduction of skew constacyclic codes of length np^s over R_k = F_{p^m}[u]/(u^k) to the study of left ideals in the quotient by f(x)^j, where f is a central irreducible divisor of x^{np^s} - λ. They follow this with explicit work on lengths 3p^s and a case-by-case treatment of skew cyclic and negacyclic codes for length 6p^s, plus some examples with good parameters. The isomorphism they set up between skew cyclic and constacyclic codes when λ sits in the base field and Θ meets suitable conditions is a direct extension of earlier results on skew polynomials over these rings. The approach uses the standard left ideal description in the quotient ring, which keeps things manageable for the nilpotent case. That part is solid for the lengths they handle in detail. The soft spot is the starting assumption that such a central irreducible f(x) of degree l and multiplicity j exists and that the quotient captures the full ideal lattice of the codes. Centrality under the automorphism Θ does not hold automatically for arbitrary Θ or general n coprime to p, and the abstract presents it as given rather than proving existence or supplying a factorization method that works broadly. If the full proofs do not close this gap, the classifications apply only where the factor is already known to exist, which narrows the reach. The paper stays within algebraic coding theory and engages the literature on skew constacyclic codes without obvious circularity or fitting issues. It is aimed at specialists who want concrete constructions and ideal structures for these specific lengths and rings. A reader focused on explicit generator polynomials or parameter tables for small p^s would get direct value. The thinking is clear enough on its own terms to merit a serious referee who can check the proofs of the isomorphisms and the scope of the reduction. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a unified approach to the study of skew constacyclic codes of length np^s over the chain ring R_k = F_{p^m}[u]/<u^k> (gcd(n,p)=1) in the skew polynomial ring R_k[x; Θ]. It reduces these codes to skew polycyclic codes of length jl associated with a central irreducible divisor f(x)^j of x^{np^s} - λ, classifies left ideals in the quotient R_k[x; Θ]/<f(x)^j> for specific Θ when λ lies in the base field F_{p^m}, establishes an isomorphism between skew cyclic and skew constacyclic codes under suitable conditions, and supplies a complete case-by-case analysis for lengths 3p^s and 6p^s together with examples of codes attaining optimal parameters.

Significance. If the reductions and classifications hold, the work supplies a systematic framework for constructing and classifying skew constacyclic codes over finite chain rings, extending existing results on skew cyclic codes and yielding explicit isomorphisms and ideal lattices that may aid the design of codes with good distance properties. The concrete analyses for small multipliers (3 and 6) and the reported optimal-parameter examples add immediate utility.

major comments (2)

[Abstract] Abstract (paragraph on reduction to polycyclic codes): The central reduction of skew constacyclic codes of length np^s to left ideals in R_k[x; Θ]/<f(x)^j> presupposes the existence of a central irreducible divisor f(x) of degree l and multiplicity j. While the text restricts to specific Θ and λ ∈ F_{p^m}, no explicit existence theorem, construction, or factorization algorithm is indicated that guarantees such an f for arbitrary n coprime to p; this assumption is load-bearing for all subsequent classification and isomorphism statements.
[Analysis for length 6p^s] Section on length 6p^s (factorization of x^{6p^s} - 1 and x^{6p^s} + 1): The case-by-case analysis must verify that each factor remains central under the chosen automorphism Θ so that the left ideal lattice of the quotient precisely captures the constacyclic codes; without this verification the isomorphism claims for negacyclic codes may not hold uniformly.

minor comments (2)

[Introduction] The phrase 'suitable conditions on Θ' appears in the abstract and introduction; these conditions should be stated explicitly (e.g., as a numbered list or proposition) at the first occurrence to improve readability.
Notation for the automorphism Θ and the ring R_k should be introduced with a single consistent definition before the first use of the skew polynomial ring.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of our work and for the detailed comments that will help improve the clarity of the manuscript. We respond to each major comment below.

read point-by-point responses

Referee: The central reduction of skew constacyclic codes of length np^s to left ideals in R_k[x; Θ]/<f(x)^j> presupposes the existence of a central irreducible divisor f(x) of degree l and multiplicity j. While the text restricts to specific Θ and λ ∈ F_{p^m}, no explicit existence theorem, construction, or factorization algorithm is indicated that guarantees such an f for arbitrary n coprime to p; this assumption is load-bearing for all subsequent classification and isomorphism statements.

Authors: We acknowledge that the reduction relies on the existence of such central irreducible divisors. The manuscript develops a general framework for when these divisors exist and then provides explicit analyses for the cases n=3 and n=6, where the factorizations of x^{np^s} - λ are explicitly constructed under the chosen automorphisms. For arbitrary n, a complete factorization algorithm is not provided as it is beyond the scope and would depend on the specific n and Θ. We will revise the abstract to clarify that the approach applies to lengths where such central divisors can be identified, and add references to existing results on skew polynomial factorizations over finite chain rings to support the assumption in the specific cases considered. revision: partial
Referee: The case-by-case analysis must verify that each factor remains central under the chosen automorphism Θ so that the left ideal lattice of the quotient precisely captures the constacyclic codes; without this verification the isomorphism claims for negacyclic codes may not hold uniformly.

Authors: We agree with the need for explicit verification. In the section on length 6p^s, the automorphisms Θ are chosen so that they are compatible with the coefficients in the factorization of x^{6p^s} ± 1, making the factors central. To make this transparent, we will add a lemma or proposition in the revised version that explicitly shows for each irreducible factor g(x) in the factorization, that Θ(g(a)) = g(Θ(a)) or the appropriate commutation relation holds, ensuring centrality in R_k[x; Θ]. This will confirm that the left ideals correspond precisely to the skew constacyclic codes. revision: yes

Circularity Check

0 steps flagged

No circularity: reduction to polycyclic codes is a standard methodological assumption, not a self-referential derivation

full rationale

The paper posits the existence of a central irreducible divisor f(x) of x^{np^s} - λ and states that the constacyclic codes reduce to left ideals in the quotient R_k[x;Θ]/⟨f(x)^j⟩. This is presented as a definitional setup for the class under study (with the assumption made explicit for specific Θ and λ in F_{p^m}), followed by direct classification of ideals and isomorphisms. No step equates a derived quantity to a fitted parameter or prior self-citation by construction; the ideal-lattice description is invoked as an external ring-theoretic fact rather than re-derived from the paper's own outputs. The case-by-case analysis for lengths 3p^s and 6p^s proceeds from explicit factorizations without looping back to the initial reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard facts from ring theory and skew polynomial rings; no new entities are postulated and no numerical parameters are fitted to data.

axioms (2)

standard math The skew polynomial ring R_k[x; Θ] is defined by the relation xa = Θ(a)x for an automorphism Θ of R_k.
Invoked in the opening setup of the skew polynomial ring.
domain assumption Central irreducible divisors of x^{np^s} - λ exist in R_k[x; Θ] and determine the ideal structure of the quotient.
Used to reduce the code problem to the study of left ideals in the quotient by f(x)^j.

pith-pipeline@v0.9.0 · 5984 in / 1490 out tokens · 39938 ms · 2026-05-19T19:33:06.991335+00:00 · methodology

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 =

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, TITLE =

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

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and Tang, X

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

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and Hesari, R.M.and Rezaei, H

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

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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 =

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and Cannon, J

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

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Antil, G

S. Antil, G. Kaur, and M. Khan, Cyclic codes of length n over finite chain rings, 2025, (preprint), arXiv:2411.16563 [math.GR]

work page arXiv 2025

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Bosma, J

W. Bosma, J. Cannon, and C. Playoust, The magma algebra system I : The user language , Journal of Symbolic Computation 24 (1997), no. 3-4, 235--265

work page 1997

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Boucher, W

D. Boucher, W. Geiselmann, and F. Ulmer, Skew-cyclic codes, Appl. Algebra Engrg. Comm. Comput. 18 (2007), no. 4, 379--389

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Bagheri, H

S. Bagheri, H. Hesari, R.M.and Rezaei, and K. Samei, Skew cyclic codes of length p^s over F_ p^m + u F_ p^m , Iran. J. Sci. Technol. Trans. A Sci. 46 (2022), no. 5, 1469--1475

work page 2022

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Skew polycyclic over finite chain rings associated to trinomials

M. Bajalan, E. Martínez-Moro, and H. Ou-azzou, Skew polycyclic over finite chain rings associated to trinomials, 2026, (preprint), arXiv:2605.03164 [math.GR]

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Y. Cao, Y. Cao, H.Q. Dinh, F.W. Fu, Jian Gao, and Songsak Sriboonchitta, Constacyclic codes of length np^s over F_ p^m +u F_ p^m , Adv. Math. Commun. 12 (2018), no. 2, 231--262

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