The aim of this paper is to determine the algebraic structure of multidimensional cyclic codes over a finite chain ring $\mathfrak{R}$. An algorithm to find the generator polynomials of $n$ dimensional ($n$D) cyclic codes of length $m_{1}m_{2}\dots m_{n}$ over $\mathfrak{R}$ has been developed using the generator polynomials of cyclic codes over $\mathfrak{R}$. Additionally, the generators of $n$D cyclic codes with length $m_{1}m_{2}\dots m_{n}$ over $\mathfrak{R}$ have been obtained as separable polynomials for the case $q\equiv 1(mod~ m_{j}), j\geq 2$, where $q=p^{r}$ is the cardinality of residue field of $\mathfrak{R}$.

翻译：本文旨在确定有限链环$\mathfrak{R}$上多维循环码的代数结构。利用$\mathfrak{R}$上循环码的生成多项式，我们提出了一种计算长度为$m_{1}m_{2}\dots m_{n}$的$n$维（$n$D）循环码生成多项式的算法。此外，针对$q\equiv 1(mod~ m_{j}), j\geq 2$的情况（其中$q=p^{r}$为$\mathfrak{R}$的剩余域的基数），我们得到了$\mathfrak{R}$上长度为$m_{1}m_{2}\dots m_{n}$的$n$D循环码的可分多项式形式的生成元。