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{$u_{l_1}\begin{pmatrix} 1\\ 0 \end{pmatrix}$, $u_{l_2}\begin{pmatrix} 0\\1 \end{pmatrix}$, $u_{l_3}\begin{pmatrix} 1\\u_1 \end{pmatrix}$, $u_{l_4}\begin{pmatrix} 1\\u_2 \end{pmatrix}$,...,$u_{l_{p^n-p^{n-1}+2}} \begin{pmatrix} 1\\u_{p^n-p^{n-1}} \end{pmatrix}$, $u_{l_{p^n-p^{n-1}+3}}\begin{pmatrix} d_1 \\ 1 \end{pmatrix}$, $u_{l_{p^n-p^{n-1}+4}}\begin{pmatrix} d_2\\ 1 \end{pmatrix}$,..., $u_{l_{p^n+1}}\begin{pmatrix} d_{p^{n-1}-1}\\1 \end{pmatrix}$, $u_{l_{p^n+2}}\begin{pmatrix} 1\\d_1 \end{pmatrix}$, $u_{l_{p^n+3}}\begin{pmatrix} 1\\d_2 \end{pmatrix}$,...,$u_{l_{p^n+p^{n-1}}}\begin{pmatrix} 1 \\d_{p^{n-1}-1} \end{pmatrix}$}}, 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：本文研究环$\mathbb{Z}_{p^n}$（其中$p$为素数）上的二维极小线性码。我们证明：若二维线性码$M$的生成矩阵$G$包含如下类型的$p^n+p^{n-1}$个列向量 {\scriptsize{$u_{l_1}\begin{pmatrix} 1\\ 0 \end{pmatrix}$, $u_{l_2}\begin{pmatrix} 0\\1 \end{pmatrix}$, $u_{l_3}\begin{pmatrix} 1\\u_1 \end{pmatrix}$, $u_{l_4}\begin{pmatrix} 1\\u_2 \end{pmatrix}$,...,$u_{l_{p^n-p^{n-1}+2}} \begin{pmatrix} 1\\u_{p^n-p^{n-1}} \end{pmatrix}$, $u_{l_{p^n-p^{n-1}+3}}\begin{pmatrix} d_1 \\ 1 \end{pmatrix}$, $u_{l_{p^n-p^{n-1}+4}}\begin{pmatrix} d_2\\ 1 \end{pmatrix}$,..., $u_{l_{p^n+1}}\begin{pmatrix} d_{p^{n-1}-1}\\1 \end{pmatrix}$, $u_{l_{p^n+2}}\begin{pmatrix} 1\\d_1 \end{pmatrix}$, $u_{l_{p^n+3}}\begin{pmatrix} 1\\d_2 \end{pmatrix}$,...,$u_{l_{p^n+p^{n-1}}}\begin{pmatrix} 1 \\d_{p^{n-1}-1} \end{pmatrix}$}}，其中对于$1\leq i \leq p^n+p^{n-1}$、$1\leq j \leq p^{n-1}-1$，$u_i$和$d_j$分别为环$\mathbb{Z}_{p^n}$中的不同单位元与零因子，且$u_{l_i}$表示$\mathbb{Z}_{p^n}$中的单位元，则$G$生成的模为极小线性码。此外，我们证明若上述任意一类列向量未完整出现在$G$中，则生成的模不为极小线性码。