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

Abdelghaffar Chibloun, Hassan Ou-azzou, Edgar Martínez-Moro, Mustapha Najmeddine

Abstract

In this paper, we investigate polycyclic codes associated with a trinomial of arbitrary degree $n$ over a finite chain ring $ R.$ We extend the concepts of $ n $-isometry and $ n $-equivalence known for constacyclic codes to this class of codes, providing a broader framework for their structural analysis. We describe the classes of $n$-equivalence and compute their number, significantly reducing the study of trinomial codes over $R$. Additionally, we examine the special case of trinomials of the form $ x^n - a_1x - a_0 \in R[x] $ and analyze their implications. Finally, we consider the extension of our results to certain trinomial additive codes over $ R.$

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

Abstract

In this paper, we investigate polycyclic codes associated with a trinomial of arbitrary degree over a finite chain ring We extend the concepts of -isometry and -equivalence known for constacyclic codes to this class of codes, providing a broader framework for their structural analysis. We describe the classes of -equivalence and compute their number, significantly reducing the study of trinomial codes over . Additionally, we examine the special case of trinomials of the form and analyze their implications. Finally, we consider the extension of our results to certain trinomial additive codes over

Paper Structure

This paper contains 14 sections, 36 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Finite chain rings
  4. Strong Gröbner bases over finite chain rings
  5. Polycyclic codes over finite chain rings
  6. Trinomial codes
  7. Equivalence of trinomial codes
  8. Isometry between trinomial codes
  9. Equivalence between trinomial codes
  10. Number of $n$-equivalence classes
  11. Trinomials of the form $x^n-a_1x-a_0$
  12. Equivalence of some additive trinomial codes
  13. Restriction to the Teichmüller set
  14. Restriction to other subgroups of $R^\times$

Key Result

Theorem 1

(mcdonald1974finite) A finite chain ring $R$ with invariants $(p,m,r,e,t)$, is of the form where $g(u) \in \mathrm{GR}(p^{m},r)[u]$ is an Eisenstein polynomial of degree $e$, i.e. $g(u)=u^{e}-p(a_{e-1}u^{e-1}+\ldots+a_{0})$, with $a_{0}\in \mathrm{GR}(p^{m},r)^{\times}$, the set of units of $\mathrm{GR}(p^{m},r)$.

Theorems & Definitions (72)

  • Theorem 1
  • Theorem 2
  • Proposition 1
  • Proposition 2
  • Theorem 3
  • Proposition 3
  • Definition 1: Generating Set in Standard Form
  • Lemma 1
  • Lemma 2
  • proof
  • ...and 62 more