  The main objective of this paper is to extend the previously defined code
family over the ring $\mathfrak{R}=\sum\limits_{s=0}^{4} v_{5}^{s}
\mathcal{A}_{4}$ to $\mathfrak{R}^{s,m}=\sum\limits_{\varsigma=1}^{m}
v_{m}^{\varsigma-1}\mathcal{A}_{m-1}$, 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.
