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

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