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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2503.03421 |
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| _version_ | 1866909525519368192 |
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| author | Sarkar, Apurba Hansda, Kalyan Maji, Makhan |
| author_facet | Sarkar, Apurba Hansda, Kalyan Maji, Makhan |
| contents | In this paper, we study the unit graph $ G(\mathbb{Z}_n) $, where $ n $ is of the form $n = p_1^{n_1} p_2^{n_2} \dots p_r^{n_r}$, with $ p_1, p_2, \dots, p_r $ being distinct prime numbers and $ n_1, n_2, \dots, n_r $ being positive integers. We establish the connectivity of $ G(\mathbb{Z}_n) $, show that its diameter is at most three, and analyze its edge connectivity. Furthermore, we construct $ q $-ary linear codes from the incidence matrix of $ G(\mathbb{Z}_n) $, explicitly determining their parameters and duals. A primary contribution of this work is the resolution of two conjectures from \cite{Jain2023} concerning the structural and coding-theoretic properties of $ G(\mathbb{Z}_n) $. These results extend the study of algebraic graph structures and highlight the interplay between number theory, graph theory, and coding theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_03421 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Linear Codes Derived from the Structure of Unit Graphs Over $\mathbb{Z}_n$ Sarkar, Apurba Hansda, Kalyan Maji, Makhan Information Theory Commutative Algebra 06F25, 13M99, 94B05 In this paper, we study the unit graph $ G(\mathbb{Z}_n) $, where $ n $ is of the form $n = p_1^{n_1} p_2^{n_2} \dots p_r^{n_r}$, with $ p_1, p_2, \dots, p_r $ being distinct prime numbers and $ n_1, n_2, \dots, n_r $ being positive integers. We establish the connectivity of $ G(\mathbb{Z}_n) $, show that its diameter is at most three, and analyze its edge connectivity. Furthermore, we construct $ q $-ary linear codes from the incidence matrix of $ G(\mathbb{Z}_n) $, explicitly determining their parameters and duals. A primary contribution of this work is the resolution of two conjectures from \cite{Jain2023} concerning the structural and coding-theoretic properties of $ G(\mathbb{Z}_n) $. These results extend the study of algebraic graph structures and highlight the interplay between number theory, graph theory, and coding theory. |
| title | Linear Codes Derived from the Structure of Unit Graphs Over $\mathbb{Z}_n$ |
| topic | Information Theory Commutative Algebra 06F25, 13M99, 94B05 |
| url | https://arxiv.org/abs/2503.03421 |