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

Abstract

In this paper, we develop the python code for generating unit graph $G({\mathbb{Z}}_n\oplus {\mathbb{Z}}_m)$, for any integers $m\&n$. For any prime $r$, we construct $r$-ary linear codes from the incidence matrix of the unit graph $G({\mathbb{Z}}_n\oplus {\mathbb{Z}}_m)$, where $n\&m$ are either power of prime or product of power of primes. We also prove the minimum distance of dual of the constructed codes as either 3 or 4. Finally, we state conjectures two on linear codes constructed from the unit graph $G({\mathbb{Z}}_n\oplus {\mathbb{Z}}_m)$, for any integer $m\&n$.