A linear code of length $n$ over a finite chain ring $R$ with residue field $\F_q$ is a $R$-submodule of $R^n$. A $R$-linear code is a code over $\F_q$ (not necessarily linear) which is the generalized Gray map image of a linear code over $R$. These codes can be seen as a generalization of the linear codes over $\Z_{p^s}$ with $p$ prime and $s \geq 1$. In this paper, we present the construction of linear simplex codes over $R$ and their corresponding $R$-linear simplex codes of type $α$ and $β$. Moreover, we show the fundamental parameters of these codes, including their minimum Hamming distance, as well as their complete weight distributions. We also study whether these simplex codes are optimal with respect to the Griesmer-type bound.

翻译：有限链环$R$上长度为$n$的线性码是$R^n$的一个$R$-子模，其中$R$的剩余域为$\F_q$。$R$-线性码是指$\F_q$上的一个码（未必是线性的），它是$R$上线性码的广义格雷映射像。这类码可视为素数$p$与$s \geq 1$时$\Z_{p^s}$上线性码的推广。本文给出了$R$上线性单纯形码及其对应的$α$型和$β$型$R$-线性单纯形码的构造，并展示了这些码的基本参数，包括其最小汉明距离及其完全权分布。此外，我们还研究了这些单纯形码关于Griesmer型界的最优性。