A Gr\"obner Approach to Dual-Containing Cyclic Left Module $(\theta,\delta)$-Codes over Finite Commutative Frobenius Rings

TL;DR

This paper develops a Gr"obner basis approach to analyze dual-containing cyclic left module codes over finite commutative Frobenius rings, providing explicit parity check matrices and algorithms for dual code verification.

Contribution

It introduces a method to compute parity check matrices and dual codes for cyclic left module codes over Frobenius rings using Gr"obner bases, extending coding theory tools.

Findings

Derived parity check matrices for these codes.

Provided algorithms to verify dual code properties.

Illustrated methods with examples over rings of order 4 and Galois rings.

Abstract

For a skew polynomial ring $R=A[X;\theta,\delta]$ where $A$ is a commutative Frobenius ring, $\theta$ an endomorphism of $A$ and $\delta$ a $\theta$-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 $\delta$ and $\theta$ to $B$ are polynomial maps. If a Gr\"obner 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 Gr\"obner 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 $4$.