pith. sign in

arxiv: 2605.03164 · v1 · submitted 2026-05-04 · 💻 cs.IT · math.IT

Skew polycyclic over finite chain rings associated to trinomials

Maryam Bajalan , Edgar Mart\'inez-Moro , Hassan Ou-azzou This is my paper

Pith reviewed 2026-05-08 17:14 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords skew polycyclic codesfinite chain ringsHamming equivalencecentral trinomialsskew polynomialsSchur multiplicationequivalence classes
0
0 comments X

The pith

Skew polycyclic codes over finite chain rings defined by central trinomials are Hamming equivalent to a canonical form under determined conditions.

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

The paper shows that an equivalence relation can be defined on central trinomials that define skew polycyclic codes over finite chain rings. This relation has a group-theoretic description using binomials under Schur multiplication. Under this relation, many such codes turn out to be Hamming equivalent to the specific codes defined by the trinomial x^n - (x^ℓ + 1). This equivalence reduces the problem of classifying these codes to a single canonical case. The size of each equivalence class is found by decomposing the unit group of the chain ring.

Core claim

We introduce an equivalence relation on the defining central trinomials for skew polycyclic codes over finite chain rings. This relation admits a group-theoretic characterization in terms of a group of binomials equipped with the Schur multiplication. We determine the conditions under which the codes are Hamming equivalent to those defined by x^n-(x^ℓ+1), reducing the classification up to Hamming equivalence to this canonical case. We also determine the size of the equivalence classes using the decomposition of the unit group of the underlying chain ring.

What carries the argument

The equivalence relation on defining trinomials characterized group-theoretically by binomials with Schur multiplication.

If this is right

  • Classification of skew polycyclic codes up to Hamming equivalence reduces to studying the canonical trinomial x^n-(x^ℓ+1).
  • The equivalence classes of trinomials are determined by the action of the group of binomials.
  • The size of these classes is computed from the unit group of the finite chain ring.
  • Conditions for Hamming equivalence in the skew setting are explicitly given for central trinomials.

Where Pith is reading between the lines

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

  • Similar equivalence relations might be definable for other classes of polynomials defining codes over rings.
  • This could facilitate the construction of new families of equivalent codes with desired properties.
  • Connections to automorphism groups in coding theory may emerge from the group-theoretic characterization.
  • Practical algorithms for checking Hamming equivalence could be developed based on this reduction.

Load-bearing premise

The trinomials are central elements in the skew polynomial ring over the finite chain ring, which permits the equivalence to be characterized via the group of binomials with Schur multiplication.

What would settle it

A specific central trinomial not satisfying the conditions for equivalence to x^n-(x^ℓ+1) whose associated skew polycyclic code is nonetheless Hamming equivalent to one from that trinomial would falsify the determined conditions.

read the original abstract

This work studies skew polycyclic codes over finite chain rings defined by central trinomials. For this class of codes, we investigate Hamming equivalence in the non-commutative (skew) setting. We introduce an equivalence relation on the defining trinomials and demonstrate that it admits a group-theoretic characterization in terms of a group of binomials equipped with the Schur multiplication. We determine the conditions under which skew polycyclic codes are Hamming equivalent to those defined by the specific trinomial $x^n-(x^\ell+1)$. This reduces the classification problem for these codes, up to Hamming equivalence, to a canonical case. Finally, we determine the size of the corresponding equivalence class using the decomposition of the unit group of the underlying chain ring.

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.

Referee Report

0 major / 3 minor

Summary. The paper studies skew polycyclic codes over finite chain rings defined by central trinomials. It introduces an equivalence relation on the defining trinomials that admits a group-theoretic characterization in terms of binomials with Schur multiplication. The authors determine the conditions under which these codes are Hamming equivalent to those defined by the trinomial x^n - (x^ℓ + 1), reducing the classification problem to a canonical case, and compute the size of the equivalence classes using the decomposition of the unit group of the chain ring.

Significance. If the derivations hold, the work offers a structured reduction of the classification problem for skew polycyclic codes up to Hamming equivalence by mapping to a canonical trinomial form. The group-theoretic characterization via binomials under Schur multiplication is a clear strength, as is the explicit computation of class sizes from the unit group; these elements make the results potentially useful for further enumeration and property analysis in non-commutative coding theory over finite chain rings.

minor comments (3)
  1. The abstract and introduction would benefit from a brief inline definition or reference for 'Schur multiplication' on binomials to ensure accessibility for readers outside the immediate subfield.
  2. Consider adding a small concrete example (e.g., for a small n and specific chain ring) that illustrates the equivalence relation and the reduction to the canonical form x^n - (x^ℓ + 1).
  3. Notation for the skew polynomial ring and centrality condition should be uniformly introduced in the preliminaries section before first use in the main theorems.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The referee's description accurately captures the paper's focus on Hamming equivalence of skew polycyclic codes over finite chain rings via an equivalence relation on central trinomials, its group-theoretic characterization under Schur multiplication, and the reduction to the canonical form x^n - (x^ℓ + 1).

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation begins by defining an equivalence relation on central trinomials in the skew polynomial ring, then establishes its group-theoretic characterization via the group of binomials under Schur multiplication. From this, the paper derives explicit conditions for Hamming equivalence to the canonical trinomial x^n-(x^ℓ+1) and computes the size of each equivalence class via the unit group decomposition of the chain ring. All steps rely on standard external algebraic machinery (skew polynomial rings, finite chain rings, group actions) rather than any self-definition, fitted parameter renamed as prediction, or load-bearing self-citation. The centrality assumption is stated as part of the setup, not smuggled in. The classification reduction is therefore a genuine consequence of the introduced relation and not equivalent to the inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard properties of finite chain rings and skew polynomial rings being central; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Finite chain rings admit skew polynomial rings with central trinomials
    Invoked implicitly when defining the codes and equivalence
  • domain assumption The unit group of the chain ring decomposes in a usable way for counting equivalence classes
    Used to determine the size of equivalence classes

pith-pipeline@v0.9.0 · 5424 in / 1286 out tokens · 24973 ms · 2026-05-08T17:14:38.322661+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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

    cs.IT 2026-05 unverdicted novelty 5.0

    The paper classifies left ideals in skew polynomial rings to describe skew constacyclic codes of length np^s over R_k and provides explicit analyses for lengths 3p^s and 6p^s with examples of optimal parameters.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    On the duality and the direction of polycyclic codes

    Adel Alahmadi, Steven Dougherty, Andr´ e Leroy, and Patrick Sol´ e. “On the duality and the direction of polycyclic codes”. In:Advances in Mathematics of Communications 10.4 (2016), pp. 921–929

    work page 2016

  2. [2]

    Polycyclic codes associated with tri- nomials: good codes and open questions

    Nuh Aydin, Peihan Liu, and Bryan Yoshino. “Polycyclic codes associated with tri- nomials: good codes and open questions”. In:Designs, Codes and Cryptography90.5 (2022), pp. 1241–1269

    work page 2022

  3. [3]

    Equivalence of Famillies of Polycyclic Codes over Finite Fields

    Hassan Ou-azzou and Anna-Lena Horlemann. “Equivalence of Famillies of Polycyclic Codes over Finite Fields”. In:https://arxiv.org/abs/2503.04498v1(2025)

    work page internal anchor Pith review arXiv 2025

  4. [4]

    On isometry and equiva- lence of skew constacyclic codes

    Hassan Ou-azzou, Mustapha Najmeddine, and Nuh Aydin. “On isometry and equiva- lence of skew constacyclic codes”. In:Discrete Mathematics348.1 (2025), p. 114279

    work page 2025

  5. [5]

    (σ, δ)-polycyclic codes in Ore extensions over rings

    Maryam Bajalan, Ivan Landjev, Edgar Mart´ ınez-Moro, and Steve Szabo. “(σ, δ)-polycyclic codes in Ore extensions over rings”. In:arXiv preprint arXiv:2312.07193(2023)

    work page arXiv 2023

  6. [6]

    On the structure of repeated-root polycyclic codes over local rings

    Maryam Bajalan, Edgar Mart´ ınez-Moro, Reza Sobhani, Steve Szabo, and G¨ uls¨ um G¨ ozde Yılmazg¨ u¸ c. “On the structure of repeated-root polycyclic codes over local rings”. In:Discrete Mathematics347.1 (2024), Paper No. 113715, 13

    work page 2024

  7. [7]

    A transform approach to polycyclic and serial codes over rings

    Maryam Bajalan, Edgar Mart´ ınez-Moro, and Steve Szabo. “A transform approach to polycyclic and serial codes over rings”. In:Finite Fields and their Applications80 (2022), Paper No. 102014, 18

    work page 2022

  8. [8]

    Jos´ e Luis Bueso, Jos´ e G´ omez-Torrecillas, and Alain Verschoren.Algorithmic methods in non-commutative algebra: Applications to quantum groups. Vol. 17. Springer Science & Business Media, 2013

    work page 2013

  9. [9]

    Repeated-root constacyclic codes of length lps and their duals

    Bocong Chen, Hai Q. Dinh, and Hongwei Liu. “Repeated-root constacyclic codes of length lps and their duals”. In:Discrete Applied Mathematics177 (2014), pp. 60–70

    work page 2014

  10. [10]

    Constacyclic codes over finite fields

    Bocong Chen, Yun Fan, Liren Lin, and Hongwei Liu. “Constacyclic codes over finite fields”. In:Finite Fields and Their Applications18.6 (2012), pp. 1217–1231. 21

    work page 2012

  11. [11]

    On polycyclic linear and additive codes associated to a trinomial over a finite chain ring

    Abdelghaffar Chibloun, Hassan Ou-azzou, Edgar Mart´ ınez-Moro, and Mustapha Na- jmeddine. “On polycyclic linear and additive codes associated to a trinomial over a finite chain ring”. In:arXiv preprint arXiv:2503.11765(2025)

    work page arXiv 2025

  12. [12]

    On isometry and equivalence of constacyclic codes over finite chain rings

    Abdelghaffar Chibloun, Hassan Ou-azzou, and Mustapha Najmeddine. “On isometry and equivalence of constacyclic codes over finite chain rings”. In:Cryptography and Communications(2024), pp. 1–25

    work page 2024

  13. [13]

    Enumeration of finite commutative chain rings

    W Edwin Clark and Joseph J Liang. “Enumeration of finite commutative chain rings”. In:Journal of Algebra27.3 (1973), pp. 445–453

    work page 1973

  14. [14]

    On poly- cyclic codes over a finite chain ring

    Alexandre Fotue-Tabue, Edgar Mart´ ınez-Moro, and J Thomas Blackford. “On poly- cyclic codes over a finite chain ring”. In:Advances in Mathematics of Communications 14.3 (2020), pp. 455–466

    work page 2020

  15. [15]

    MacWilliams’ Extension Theorem for bi-invariant weights over finite principal ideal rings

    Marcus Greferath, Thomas Honold, Cathy Mc Fadden, Jay A Wood, and Jens Zumbr¨ agel. “MacWilliams’ Extension Theorem for bi-invariant weights over finite principal ideal rings”. In:Journal of Combinatorial Theory, Series A125 (2014), pp. 177–193

    work page 2014

  16. [16]

    Vandermonde and Wronskian matrices over division rings

    Tsit-Yuen Lam and Andr´ e Leroy. “Vandermonde and Wronskian matrices over division rings”. In:Journal of Algebra119.2 (1988), pp. 308–336

    work page 1988

  17. [17]

    Homomorphisms between Ore extensions

    TY Lam and A Leroy. “Homomorphisms between Ore extensions”. In:Comtemporary Mathematics124 (1992), pp. 83–110

    work page 1992

  18. [18]

    Algebraic conjugacy classes and skew polynomial rings

    TY Lam and Andr´ e Leroy. “Algebraic conjugacy classes and skew polynomial rings”. In:Perspectives in ring theory. Springer, 1988, pp. 153–203

    work page 1988

  19. [19]

    McDonald.Finite Rings with Identity

    B.R. McDonald.Finite Rings with Identity. Lecture notes in pure and applied mathe- matics. M. Dekker, 1974

    work page 1974

  20. [20]

    Using nonassociative algebras to classify skew polycyclic codes up to isometry and equivalence

    Susanne Pumpl¨ un. “Using nonassociative algebras to classify skew polycyclic codes up to isometry and equivalence”. In:arXiv preprint arXiv:2508.10139(2025)

    work page arXiv 2025

  21. [21]

    How easy is code equivalence over Fq?

    Nicolas Sendrier and Dimitrios E Simos. “How easy is code equivalence over Fq?” In: International Workshop on Coding and Cryptography-WCC 2013. 2013

    work page 2013

  22. [22]

    Polycyclic codes as invari- ant subspaces

    Minjia Shi, Xiaoxiao Li, Zahra Sepasdar, and Patrick Sol´ e. “Polycyclic codes as invari- ant subspaces”. In:Finite Fields Appl.68 (2020), pp. 101760, 14

    work page 2020

  23. [23]

    Equivalence and duality of polycyclic codes associated with trinomials over finite fields

    Minjia Shi, Haodong Lu, Shuang Zhou, Jiarui Xu, and Yuhang Zhu. “Equivalence and duality of polycyclic codes associated with trinomials over finite fields”. In:Finite Fields and Their Applications91 (2023), p. 102259. 22

    work page 2023