Codes over Affine Algebras with a Finite Commutative Chain coefficient Ring

Abstract

We consider codes defined over an affine algebra $\mathcal{A}=R[X_1,\dots ,X_r]/⟨t_1(X_1),\dots ,t_r(X_r)⟩$, 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.

Submitted to Finite Fields and Their Applications