In this paper we study two dimensional minimal linear code over the ring $\mathbb{Z}_{p^n}$(where $p$ is prime). We show that if the generator matrix $G$ of the two dimensional linear code $M$ contains $p^n+p^{n-1}$ column vector of the following type {\scriptsize{$u_{l_1}\begin{pmatrix} 1\\ 0 \end{pmatrix}$, $u_{l_2}\begin{pmatrix} 0\\1 \end{pmatrix}$, $u_{l_3}\begin{pmatrix} 1\\u_1 \end{pmatrix}$, $u_{l_4}\begin{pmatrix} 1\\u_2 \end{pmatrix}$, \ldots,$u_{l_{p^n-p^{n-1}+2}} \begin{pmatrix} 1\\u_{p^n-p^{n-1}} \end{pmatrix}$, $u_{l_{p^n-p^{n-1}+3}}\begin{pmatrix} d_1 \\ 1 \end{pmatrix}$, $u_{l_{p^n-p^{n-1}+4}}\begin{pmatrix} d_2\\ 1 \end{pmatrix}$,\ldots \dots,\\ $u_{l_{p^n+1}}\begin{pmatrix} d_{p^{n-1}-1}\\1 \end{pmatrix}$, $u_{l_{p^n+2}}\begin{pmatrix} 1\\d_1 \end{pmatrix}$, $u_{l_{p^n+3}}\begin{pmatrix} 1\\d_2 \end{pmatrix}$,\ldots,$u_{l_{p^n+p^{n-1}}}\begin{pmatrix} 1 \\d_{p^{n-1}-1} \end{pmatrix}$}}, where $u_i$ and $d_j$ are distinct units and zero divisors respectively in the ring $\mathbb{Z}_{p^n}$ for $1\leq i \leq p^n+p^{n-1}$, $1\leq j \leq p^{n-1}-1$ and additionally, denote $u_{l_i}$ as units in $\mathbb{Z}_{p^n}$, then the module generated by $G$ is a minimal linear code. Also we show that if any one column vector of the above types are not present entirely in $G$, then the generated module is not a minimal linear code.

翻译：本文研究了环 $\mathbb{Z}_{p^n}$（其中 $p$ 为素数）上的二维极小线性码。我们证明了：若二维线性码 $M$ 的生成矩阵 $G$ 包含以下类型的 $p^n+p^{n-1}$ 个列向量 {\scriptsize{$u_{l_1}\begin{pmatrix} 1\\ 0 \end{pmatrix}$, $u_{l_2}\begin{pmatrix} 0\\1 \end{pmatrix}$, $u_{l_3}\begin{pmatrix} 1\\u_1 \end{pmatrix}$, $u_{l_4}\begin{pmatrix} 1\\u_2 \end{pmatrix}$, \ldots,$u_{l_{p^n-p^{n-1}+2}} \begin{pmatrix} 1\\u_{p^n-p^{n-1}} \end{pmatrix}$, $u_{l_{p^n-p^{n-1}+3}}\begin{pmatrix} d_1 \\ 1 \end{pmatrix}$, $u_{l_{p^n-p^{n-1}+4}}\begin{pmatrix} d_2\\ 1 \end{pmatrix}$,\ldots \dots,\\ $u_{l_{p^n+1}}\begin{pmatrix} d_{p^{n-1}-1}\\1 \end{pmatrix}$, $u_{l_{p^n+2}}\begin{pmatrix} 1\\d_1 \end{pmatrix}$, $u_{l_{p^n+3}}\begin{pmatrix} 1\\d_2 \end{pmatrix}$,\ldots,$u_{l_{p^n+p^{n-1}}}\begin{pmatrix} 1 \\d_{p^{n-1}-1} \end{pmatrix}$}}，其中 $u_i$ 和 $d_j$ 分别为环 $\mathbb{Z}_{p^n}$ 中互异的单位元和零因子，$1\leq i \leq p^n+p^{n-1}$, $1\leq j \leq p^{n-1}-1$，且记 $u_{l_i}$ 为 $\mathbb{Z}_{p^n}$ 中的单位元，则 $G$ 生成的模是一个极小线性码。同时，我们还证明了：若上述类型中的任意一个列向量未完整出现在 $G$ 中，则生成的模不是极小线性码。