Construction of Linear Codes from the Unit Graph \(G(\mathbb{Z}_{n}\oplus \mathbb{Z}_{m})\)

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 \(m\ \& \ n\).