Construction of Linear Codes from the Unit Graph $G(\mathbb{Z}_{n})$
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| Format: | Preprint |
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2023
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| _version_ | 1866913485616578560 |
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| author | Jain, Rupali S. Reddy, B. Surendranath Shaikh, Wajid M. |
| author_facet | Jain, Rupali S. Reddy, B. Surendranath Shaikh, Wajid M. |
| contents | In this paper, we consider the unit graph $G(\mathbb{Z}_{n})$, where $n=p_{1}^{n_{1}} \text{ or } p_{1}^{n_{1}}p_{2}^{n_{2}} \text{ or } p_{1}^{n_{1}}p_{2}^{n_{2}}p_{3}^{n_{3}}$ and $p_{1}, p_{2}, p_{3}$ are distinct primes. For any prime $q$, we construct $q$-ary linear codes from the incidence matrix of the unit graph $G(\mathbb{Z}_{n})$ with their parameters. We also prove that the dual of the constructed codes have minimum distance either 3 or 4. Lastly, we stated two conjectures on diameter of unit graph $G(\mathbb{Z}_{n})$ and linear codes constructed from the incidence matrix of the unit graph $G(\mathbb{Z}_{n})$ for any integer $n$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_05169 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Construction of Linear Codes from the Unit Graph $G(\mathbb{Z}_{n})$ Jain, Rupali S. Reddy, B. Surendranath Shaikh, Wajid M. Rings and Algebras Combinatorics 94B05, 05C50, 05C38 In this paper, we consider the unit graph $G(\mathbb{Z}_{n})$, where $n=p_{1}^{n_{1}} \text{ or } p_{1}^{n_{1}}p_{2}^{n_{2}} \text{ or } p_{1}^{n_{1}}p_{2}^{n_{2}}p_{3}^{n_{3}}$ and $p_{1}, p_{2}, p_{3}$ are distinct primes. For any prime $q$, we construct $q$-ary linear codes from the incidence matrix of the unit graph $G(\mathbb{Z}_{n})$ with their parameters. We also prove that the dual of the constructed codes have minimum distance either 3 or 4. Lastly, we stated two conjectures on diameter of unit graph $G(\mathbb{Z}_{n})$ and linear codes constructed from the incidence matrix of the unit graph $G(\mathbb{Z}_{n})$ for any integer $n$. |
| title | Construction of Linear Codes from the Unit Graph $G(\mathbb{Z}_{n})$ |
| topic | Rings and Algebras Combinatorics 94B05, 05C50, 05C38 |
| url | https://arxiv.org/abs/2307.05169 |