Codes over Affine Algebras with a Finite Commutative Chain coefficient Ring

TL;DR

This paper studies a new class of algebraic codes over affine algebras built from finite commutative chain rings, generalizing existing codes and exploring their structure and duality properties.

Contribution

It introduces a canonical generator matrix for these codes and extends duality concepts to this new algebraic setting.

Findings

Codes generalize polynomial quotient and multivariable codes

Canonical generator matrices are constructed for these codes

Duality properties are established for the codes

Abstract

We consider codes defined over an affine algebra $\mathcal A=R[X_1,\dots,X_r]/\left\langle t_1(X_1),\dots,t_r(X_r)\right\rangle$, where $t_i(X_i)$ is a monic univariate polynomial over a finite commutative chain ring $R$. Namely, we study the $\mathcal A-$submodules of $\mathcal A^l$ ($l\in \mathbb{N}$). These codes generalize both the codes over finite quotients of polynomial rings and the multivariable codes over finite chain rings. {Some codes over Frobenius local rings that are not chain rings are also of this type}. A canonical generator matrix for these codes is introduced with the help of the Canonical Generating System. Duality of the codes is also considered.