Construction of Linear Codes from the Unit Graph $G({\mathbb{Z}}_n)$

Abstract

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 $n$.