Generalization the parameters of minimal linear codes over the ring $\mathbbZ_p^l$ and $\mathbbZ_p_1p_2$
<|reference_start|>Generalization the parameters of minimal linear codes over the ring $\mathbbZ_p^l$ and $\mathbbZ_p_1p_2$: 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 $m\geq m(k;q)$ 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 $m(k;p_1p_2)$ within the ring $\mathbb{Z}_{p^l}$ and $\mathbb{Z}_{p_1p_2}$ respectively.<|reference_end|>
@article{chatterjee2024generalization,
    title={Generalization the parameters of minimal linear codes over the ring
  $\mathbb{Z}_{p^l}$ and $\mathbb{Z}_{{p_1}{p_2}}$},
    author={Biplab Chatterjee and Ratnesh Kumar Mishra},
    journal={arXiv preprint arXiv:2404.09561},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.09561},
    primaryClass={cs.IT math.IT math.RA}
}