arxiv: 2605.13020 · v1 · submitted 2026-05-13 · 💻 cs.IT · math.IT

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

Akanksha Tiwari , Ritumoni Sarma

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Pith reviewed 2026-05-14 18:25 UTC · model grok-4.3

classification 💻 cs.IT math.IT

keywords skew polycyclic codesleft idealschain ringskew polynomial ringconstacyclic codestorsion codes

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The pith

The structure of left ideals in skew polynomial rings over finite chain rings is refined for central elements x^{np^s} - λ with nonzero constant term.

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

This paper determines the form of left ideals in the quotient of the skew polynomial ring R^t[x, Θ] by a central polynomial f(x), where R^t is the chain ring F_{p^m}[u]/(u^t). These ideals represent skew polycyclic codes. Special attention is given to f(x) = x^{np^s} - λ when n equals 1 or 2, providing a more detailed generator description especially when the linear coefficient of λ is nonzero. For the particular values n=1 with t=3 and n=2 with t=2, a complete listing of all such left ideals is supplied after adding conditions that were missing from earlier studies, which ensures the classes remain separate and permits calculation of certain torsion codes.

Core claim

The left ideals are generated according to the valuation of the leading coefficient or similar parameters, but with explicit forms that incorporate the extension of the automorphism and the nonzero λ0. When λ1 ≠0, the generators simplify. The full description for the small cases includes all possible ideals under the refined conditions.

What carries the argument

The quotient ring R^t[x, Θ]/<f(x)> with f central and Θ the ring automorphism fixing u, whose left ideals form the skew polycyclic codes.

If this is right

The classes of left ideals are mutually disjoint when the additional necessary conditions are included.
Explicit generators are available for all left ideals in the cases n=1 t=3 and n=2 t=2.
i-th torsion codes can be determined explicitly for some of these ideals.
Simpler generator polynomials arise when the coefficient λ1 is nonzero.

Where Pith is reading between the lines

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

The refined structure may enable systematic enumeration of all skew constacyclic codes for these parameters.
Similar refinements could apply to larger values of n and t to obtain general classifications.
Knowledge of the ideal structure supports the study of minimum distances in these codes.

Load-bearing premise

The automorphism Θ of the ring R^t extends the field automorphism θ while fixing u, and the constant term λ0 in the central element λ is nonzero.

What would settle it

An explicit pair of distinct left ideals whose generators violate the added conditions and intersect nontrivially for n=1 and t=3 would show the classes are not disjoint without them.

read the original abstract

Let $R^t$ denote the finite chain ring $\frac{\mathbb{F}_{p^m}[u]}{\langle u^t \rangle},$ where $p$ is a prime and $t$ is a positive integer. In this article, for a prime $p$ and an automorphism $\theta$ of $\mathbb{F}_{p^m}$, we give the structure of the left ideals of the ring $\frac{R^t[x,\Theta]}{\langle f(x) \rangle},$ where $f(x)$ is in the center of the skew polynomial ring $R^t[x,\Theta]$ and $\Theta$ is an automorphism of $R^t$ that extends $\theta$ with $\Theta(u)=u$. These left ideals are also referred to as skew polycyclic codes associated to $f(x).$ In particular, when the central element $ f(x)$ is $x^{np^s}-\lambda $, where $\lambda=\lambda_0+u\lambda_1+\cdots +u^{t-1}\lambda_{t-1}$ with $\lambda_0\ne0,$ and $ n=1,2 $, we give a more refined form of the left ideals (which are also called skew constacyclic codes). Moreover, the case $\lambda_1 \neq 0$ is analyzed in detail, yielding a simpler form of generators that reveals a more refined structural characterization of the left ideals. As an application, for $n=1,t=3$ and $n=2,t=2$ we give a full description of the left ideals by including certain necessary conditions that were omitted in available literature, preventing the different classes of left ideals from being mutually disjoint and in certain cases, we also compute $i$-th torsion codes.

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 adds necessary conditions to separate left ideal classes in skew constacyclic codes for n=1 t=3 and n=2 t=2, with torsion code computations, but the disjointness needs explicit verification.

read the letter

The main thing to know is that this paper fills in some missing conditions for the left ideals of skew polynomial rings over chain rings to form disjoint classes, at least for the small parameters n=1 with t=3 and n=2 with t=2, and it works out the torsion codes in those settings. It builds on existing work by giving a more refined structure when the constant term λ has λ1 nonzero. The authors extend the automorphism and focus on central elements of the form x to the np^s minus λ. This leads to simpler generators in the λ1 case and a claimed complete list of ideals that avoids overlaps that earlier papers had. The work is careful with the setup, using the standard extension of the automorphism Θ that fixes u. It seems to correct an oversight in the literature about what makes the ideal classes separate. One soft spot is the lack of a direct check, like listing all possibilities for the smallest case, to confirm that the new conditions are both necessary and sufficient. The stress on the partition being load-bearing makes sense, and if the derivations don't fully rule out overlaps, the torsion code results could be affected. Still, the abstract indicates they did analyze the λ1 case in detail. This is aimed at researchers in coding theory who deal with skew polycyclic codes over rings. Someone looking for explicit structures in small cases will find value, even if broader applications aren't claimed. It is solid enough to warrant a serious referee, who can probe the verification of those conditions. I would send it out for peer review.

Referee Report

2 major / 2 minor

Summary. The paper determines the structure of left ideals (skew polycyclic codes) in the quotient ring R^t[x, Θ]/⟨f(x)⟩ where R^t = F_{p^m}[u]/(u^t) is a finite chain ring and f(x) is central in the skew polynomial ring. For the special case f(x) = x^{n p^s} - λ with n = 1,2 and λ_0 ≠ 0, it supplies refined generator forms for skew constacyclic codes, with a detailed analysis when λ_1 ≠ 0. As an application, for the parameters n=1,t=3 and n=2,t=2 it claims a complete, mutually disjoint classification of left ideals by adding necessary conditions omitted from prior literature, together with explicit computations of the i-th torsion codes.

Significance. If the added conditions indeed produce a partition into mutually disjoint classes and the torsion-code formulas are correct, the work supplies a corrected and more precise structural description of skew constacyclic codes over these chain rings. Such classifications are useful for constructing and enumerating codes with prescribed properties; the explicit torsion-code results would also enable direct computation of minimum distances and other invariants.

major comments (2)

[subsection on n=1, t=3] The subsection treating n=1, t=3: the claim that the newly supplied conditions on the generators (for λ_1 ≠ 0) are necessary to guarantee that the listed classes are mutually disjoint is load-bearing for the 'full description' assertion, yet the manuscript provides no exhaustive enumeration of all possible left ideals for the smallest admissible parameters to confirm that no ideal satisfies two distinct generator forms simultaneously.
[subsection on n=1, t=3] The same subsection: the formulas for the i-th torsion codes are derived from the refined generators; if any overlap between classes exists, these formulas become invalid for at least one ideal. No independent verification (e.g., direct computation of the torsion submodule for a concrete generator) is supplied.

minor comments (2)

[introduction] The statement that Θ extends θ with Θ(u)=u should be accompanied by an explicit verification that this extension is indeed an automorphism of R^t for every t.
[section on skew constacyclic codes] Notation for the central element λ = λ_0 + u λ_1 + … should be introduced once and used consistently; the current presentation repeats the expansion in several places.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and for identifying the need for explicit verification of the disjointness and torsion-code claims in the n=1, t=3 case. We will revise the manuscript to incorporate concrete enumerations and direct computations that address these points.

read point-by-point responses

Referee: [subsection on n=1, t=3] The subsection treating n=1, t=3: the claim that the newly supplied conditions on the generators (for λ_1 ≠ 0) are necessary to guarantee that the listed classes are mutually disjoint is load-bearing for the 'full description' assertion, yet the manuscript provides no exhaustive enumeration of all possible left ideals for the smallest admissible parameters to confirm that no ideal satisfies two distinct generator forms simultaneously.

Authors: We agree that an explicit enumeration for the smallest admissible parameters would provide concrete confirmation. Our algebraic arguments show that the added conditions on the generators force the classes to be disjoint, but we will strengthen the presentation by including, in the revised version, a complete listing of all left ideals for the smallest parameters (p=2, m=1) and verifying that no generator satisfies more than one of the listed forms simultaneously. revision: yes
Referee: [subsection on n=1, t=3] The same subsection: the formulas for the i-th torsion codes are derived from the refined generators; if any overlap between classes exists, these formulas become invalid for at least one ideal. No independent verification (e.g., direct computation of the torsion submodule for a concrete generator) is supplied.

Authors: The torsion-code formulas are obtained directly from the refined generator descriptions once disjointness is established. To supply the requested independent check, the revised manuscript will contain explicit computations of the i-th torsion submodules for one representative generator from each class in the n=1, t=3 case, confirming that the formulas match the direct calculations. revision: yes

Circularity Check

0 steps flagged

No circularity: structure derived from standard skew polynomial and chain ring facts

full rationale

The derivation proceeds from the definition of the skew polynomial ring R^t[x,Θ] modulo the central element f(x) = x^{np^s} - λ, using the automorphism extension Θ(u)=u and the nilpotency u^t=0. The refined generators for λ_1 ≠ 0 and the added necessary conditions for n=1,t=3 and n=2,t=2 are obtained by direct verification against the multiplication rule x r = Θ(r) x; these conditions are not obtained by fitting parameters to data nor by renaming prior results. No self-citation is load-bearing for the partition of left ideals, and the torsion-code computations follow from the explicit generator forms without reducing to the input assumptions by construction. The work is therefore self-contained against external benchmarks in finite ring theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the standard definition of the chain ring R^t, the existence of an automorphism Θ that fixes u, and the centrality of the chosen f(x) inside the skew polynomial ring; all are domain assumptions drawn from prior literature on skew codes.

axioms (2)

domain assumption R^t = F_{p^m}[u] / <u^t> is a finite chain ring
Standard setup invoked throughout the abstract for the coefficient ring.
domain assumption Θ is an automorphism of R^t extending θ with Θ(u) = u
Required to define the skew multiplication in R^t[x, Θ].

pith-pipeline@v0.9.0 · 5633 in / 1374 out tokens · 47730 ms · 2026-05-14T18:25:52.968815+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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