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A Gröbner Approach to Dual-Containing Cyclic Left Module $(θ,δ)$-Codes over Finite Commutative Frobenius Rings

Hedongliang Liu, Cornelia Ott, Felix Ulmer

TL;DR

This work addresses the construction and characterization of dual-containing cyclic left module $(\theta,\delta)$-codes over skew polynomial rings $R=A[X;\theta,\delta]$ with $A$ a finite commutative Frobenius ring. It develops a parity-check framework entirely within skew-polynomial algebra, reduces coefficient expressions to polynomial maps over a subring $B$ when $A=B[a_1,...,a_s]$, and employs Gröbner-basis methods to enumerate all dual-containing codes for fixed parameters $(n,k)$. An algorithm is provided to decide whether the dual of a given cyclic module code is itself cyclic, and extensive computational results across small rings (including ${\mathbb F}_2[v]/(v^2+v)$, ${\mathbb F}_2[u]/(u^2)$, ${\mathbb F}_4$, and GR$(4,2)$) illustrate when nontrivial derivations or endomorphisms yield new codes and when duals maintain cyclic-structure. The findings demonstrate the utility of Gröbner-basis techniques for code search in noncommutative settings and reveal rich dual-containing phenomena with potential applications to quantum coding and noncommutative coding theory.

Abstract

For a skew polynomial ring $R=A[X;θ,δ]$ where $A$ is a commutative Frobenius ring, $θ$ an endomorphism of $A$ and $δ$ a $θ$-derivation of $A$, we consider cyclic left module codes $\mathcal{C}=Rg/Rf\subset R/Rf$ where $g$ is a left and right divisor of $f$ in $R$. In this paper, we derive a parity check matrix when $A$ is a finite commutative Frobenius ring using only the framework of skew polynomial rings. We consider rings $A=B[a_1,\ldots,a_s]$ which are free $B$-modules where the restriction of $δ$ and $θ$ to $B$ are polynomial maps. If a Gröbner basis can be computed over $B$, then we show that all Euclidean and Hermitian dual-containing codes $\mathcal{C}=Rg/Rf\subset R/Rf$ can be computed using a Gröbner basis. We also give an algorithm to test if the dual code is again a cyclic left module code. We illustrate our approach for rings of order $4$ with non-trivial endomorphism and the Galois ring of characteristic $4$.

A Gröbner Approach to Dual-Containing Cyclic Left Module $(θ,δ)$-Codes over Finite Commutative Frobenius Rings

TL;DR

This work addresses the construction and characterization of dual-containing cyclic left module -codes over skew polynomial rings with a finite commutative Frobenius ring. It develops a parity-check framework entirely within skew-polynomial algebra, reduces coefficient expressions to polynomial maps over a subring when , and employs Gröbner-basis methods to enumerate all dual-containing codes for fixed parameters . An algorithm is provided to decide whether the dual of a given cyclic module code is itself cyclic, and extensive computational results across small rings (including , , , and GR) illustrate when nontrivial derivations or endomorphisms yield new codes and when duals maintain cyclic-structure. The findings demonstrate the utility of Gröbner-basis techniques for code search in noncommutative settings and reveal rich dual-containing phenomena with potential applications to quantum coding and noncommutative coding theory.

Abstract

For a skew polynomial ring where is a commutative Frobenius ring, an endomorphism of and a -derivation of , we consider cyclic left module codes where is a left and right divisor of in . In this paper, we derive a parity check matrix when is a finite commutative Frobenius ring using only the framework of skew polynomial rings. We consider rings which are free -modules where the restriction of and to are polynomial maps. If a Gröbner basis can be computed over , then we show that all Euclidean and Hermitian dual-containing codes can be computed using a Gröbner basis. We also give an algorithm to test if the dual code is again a cyclic left module code. We illustrate our approach for rings of order with non-trivial endomorphism and the Galois ring of characteristic .
Paper Structure (14 sections, 7 theorems, 26 equations, 11 tables, 1 algorithm)

This paper contains 14 sections, 7 theorems, 26 equations, 11 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Skew Polynomial Rings
  4. Cyclic Left Module $(\theta,\delta)$-Codes $Rg/Rf\subset R/Rf$
  5. Parity Check Matrix and Generating Matrix of the Euclidean and Hermitian Dual Code
  6. Computing all Dual-Containing $(\theta,\delta)$-Codes
  7. Polynomial Maps
  8. Computations via Gröbner Basis over $B\subset A$
  9. An Algorithm to Compute All Dual-Containing $(\theta,\delta)$-Codes
  10. Is the Dual $C^{\perp_{\sigma}}$ of a Cyclic Module $(\theta,\delta)$-Code Again a Cyclic Module $(\theta,\delta)$-Code ?
  11. Computational Results for $A={\mathbb F}_2[v]/(v^2+v)$
  12. Computational Results for $A={\mathbb F}_2[u]/(u^2)$
  13. Computational Results for $A={\mathbb F}_4$
  14. Computation Results for the Galois Ring $A=GR(4,2)$

Key Result

Proposition 1

Let $g\in R=A[X;\theta,\delta]$ of degree $n-k$ be a right divisor of a monic skew polynomial $f\in R$ of degree $n$. A cyclic left module $(\theta,\delta)$-code ${\mathcal{C}}=Rg/Rf\subset R/Rf$ has the following properties:

Theorems & Definitions (23)

  • Definition 1
  • Definition 2
  • Proposition 1
  • proof
  • Example 1
  • Example 2
  • Definition 3: Hermitian Inner Product and Hermitian Dual
  • Lemma 1
  • Lemma 2
  • proof
  • ...and 13 more