Weight distributions of simplex codes over finite chain rings and their Gray map images

A linear code of length $n$ over a finite chain ring $R$ with residue field ${\text{F}}_q$ is a $R$-submodule of $R^n$. A $R$-linear code is a code over ${\text{F}}_q$ (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 ${\mathbb{Z}}_{p^s}$ with $p$ prime and $s\ge 1$. In this paper, we present the construction of linear simplex codes over $R$ and their corresponding $R$-linear simplex codes of type $\alpha$ and $\beta$. 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.