On two-dimensional minimal linear codes over the rings $\mathbb{Z}_{p^n}$

Abstract: 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.