Generalization the parameters of minimal linear codes over the ring Zpˡ

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 Zpˡ. 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 {nine}) showed that there exists a positive integer $n(k;q)$ such that for n≥ n(k;q) a minimal linear code of length n and dimension k over a finite field Fq 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;pˡ) within the ring Zpˡ.