Generalization the parameters of minimal linear codes over the ring $\mathbb{Z}_{p^l}$ and $\mathbb{Z}_{{p_1}{p_2}}$
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909915689254912 |
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| author | Chatterjee, Biplab Mishra, Ratnesh Kumar |
| author_facet | Chatterjee, Biplab Mishra, Ratnesh Kumar |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_09561 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalization the parameters of minimal linear codes over the ring $\mathbb{Z}_{p^l}$ and $\mathbb{Z}_{{p_1}{p_2}}$ Chatterjee, Biplab Mishra, Ratnesh Kumar Information Theory Rings and Algebras 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. |
| title | Generalization the parameters of minimal linear codes over the ring $\mathbb{Z}_{p^l}$ and $\mathbb{Z}_{{p_1}{p_2}}$ |
| topic | Information Theory Rings and Algebras |
| url | https://arxiv.org/abs/2404.09561 |