Let $ \mathbb F_2[u]/ \langle u^k \rangle= \mathbb F_2+u\mathbb F_2+u^2\mathbb F_2+\cdots+u^{k-1}\mathbb F_2 ,$ where $u^k=0$ for a positive integer $k$, and $\mathcal{R}=M_4 (\mathbb F_2( u)/ \langle u^k \rangle)$ be the finite noncommutative non-chain matrix ring of order $4\times4$. This paper presents the construction of cyclic codes over the finite field $\mathbb F_{16}$ via the considered matrix ring $\mathcal{R}$. In this connection, first, we discuss the structure of the ring $\mathcal{R}$ and show that $\mathcal{R}$ is isomorphic to the ring $( \mathbb F_{16}+ v\mathbb F_{16} + v^2\mathbb F_{16} + v^3\mathbb F_{16}) + u(\mathbb F_{16} + v\mathbb F_{16} + v^2\mathbb F_{16} + v^3\mathbb F_{16}) + u^2(\mathbb F_{16} + v\mathbb F_{16} + v^2\mathbb F_{16}+ v^3\mathbb F_{16}) + \cdots + u^{k-1}(\mathbb F_{16} + v\mathbb F_{16} + v^2\mathbb F_{16} + v^3\mathbb F_{16})$ where $v^4=0, u^k=0, u^iv^j=v^ju^i$ for $i \in \{1,\dots, k-1\}$ and $j \in \{1, 2, 3\}$. Then, we establish the form of ideals of the ring $\mathcal{R}$ and related cyclic codes over $\mathcal{R}$. Further, we show that these cyclic codes can be written as the direct sums of $\mathcal{R}$-submodules of $\frac{\mathcal{R}[x]}{<x^n-1>}$, and derive the formula for the cardinality of cyclic codes over $\mathcal{R}$. Then, we consider the Euclidean and Hermitian duals of the derived cyclic codes over $\mathcal{R}$. Under the module isometry for $\mathcal{R}$, we use the Bachoc map and the Gray map, which takes a derived cyclic code over $\mathcal{R}$ to $\mathbb F_{16}$. Finally, we provide some non-trivial examples of linear codes over $\mathbb F_{16}$ with good parameters that support our derived results and compare a few codes with existing codes in the literature.

翻译：令 $ \mathbb F_2[u]/ \langle u^k \rangle= \mathbb F_2+u\mathbb F_2+u^2\mathbb F_2+\cdots+u^{k-1}\mathbb F_2 ,$ 其中 $u^k=0$，$k$ 为正整数，且 $\mathcal{R}=M_4 (\mathbb F_2( u)/ \langle u^k \rangle)$ 为 $4\times4$ 阶有限非交换非链矩阵环。本文通过所考虑的矩阵环 $\mathcal{R}$ 提出了有限域 $\mathbb F_{16}$ 上循环码的构造。为此，首先我们讨论环 $\mathcal{R}$ 的结构，并证明 $\mathcal{R}$ 同构于环 $( \mathbb F_{16}+ v\mathbb F_{16} + v^2\mathbb F_{16} + v^3\mathbb F_{16}) + u(\mathbb F_{16} + v\mathbb F_{16} + v^2\mathbb F_{16} + v^3\mathbb F_{16}) + u^2(\mathbb F_{16} + v\mathbb F_{16} + v^2\mathbb F_{16}+ v^3\mathbb F_{16}) + \cdots + u^{k-1}(\mathbb F_{16} + v\mathbb F_{16} + v^2\mathbb F_{16} + v^3\mathbb F_{16})$，其中 $v^4=0, u^k=0$，且对 $i \in \{1,\dots, k-1\}$ 和 $j \in \{1, 2, 3\}$ 有 $u^iv^j=v^ju^i$。然后，我们建立了环 $\mathcal{R}$ 的理想形式以及 $\mathcal{R}$ 上相关循环码的结构。进一步，我们证明了这些循环码可以写成 $\frac{\mathcal{R}[x]}{<x^n-1>}$ 的 $\mathcal{R}$-子模的直和，并推导了 $\mathcal{R}$ 上循环码基数的计算公式。接着，我们考虑了 $\mathcal{R}$ 上所得循环码的欧几里得对偶和厄米特对偶。在 $\mathcal{R}$ 的模等距映射下，我们利用 Bachoc 映射和 Gray 映射，将 $\mathcal{R}$ 上的一个导出循环码映射到 $\mathbb F_{16}$ 上。最后，我们提供了一些具有良好参数的 $\mathbb F_{16}$ 上线性码的非平凡例子，这些例子支持我们推导的结果，并将其中一些码与文献中已有的码进行了比较。