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.

翻译：本文的主要目的是将之前定义的代码族扩展到环 $\mathfrak{R}^{s,m}=\sum\limits_{\varsigma=1}^{m} v_{m}^{\varsigma-1}\mathcal{A}_{m-1}$ 上，并提出其在编码理论中的实现扩展框架，并从这个广义的代码族中导出附加的特性，包括循环和准循环代码的构造。此外，我们将介绍这个扩展代码族的具体应用。