Generalization the parameters of minimal linear codes over the ring ${\mathbb{Z}}_{p^l}$ and ${\mathbb{Z}}_{p_1{p}_2}$

In this article, We introduce a condition that is both necessary and sufficient for a linear code to achieve minimality when analyzed over the rings ${\mathbb{Z}}_n$.The fundamental inquiry in minimal linear codes is the existence of a $[m,k]$ minimal linear code where $k$ is less than or equal to $m$. W. Lu et al. ( see \cite{nine}) showed that there exists a positive integer $m(k;q)$ such that for $m\ge m(k;q)$ a minimal linear code of length $m$ and dimension $k$ over a finite field ${\mathbb{F}}_q$ must exist. They give the upper and lower bound of $m(k;q)$. In this manuscript, we establish both an upper and lower bound for $m(k;p^l)$ and $m(k;p_1{p}_2)$ within the ring ${\mathbb{Z}}_{p^l}$ and ${\mathbb{Z}}_{p_1{p}_2}$ respectively.