Generalization the parameters of minimal linear codes over the ring Z₏⌀

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 Z₏⌀. 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 Z₏⌀.