Linear Codes over ${\mathfrak{R}}^{s,m}=\sum _{\varsigma =1}^m{v}_m^{\varsigma -1}{\mathcal{A}}_{m-1}$, with $v_m^m=v_m$

Abstract

The main objective of this paper is to extend the previously defined code family over the ring $\mathfrak{R}=\sum _{s=0}^4{v}_5^s{\mathcal{A}}_4$ to ${\mathfrak{R}}^{s,m}=\sum _{\varsigma =1}^m{v}_m^{\varsigma -1}{\mathcal{A}}_{m-1}$, and propose an expanded framework for its implementation in coding theory, and to derive additional properties from this generalized code family, including the construction of cyclic and quasi-cyclic codes. Furthermore, we will present specific applications of this extended code family.