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

TL;DR

This paper constructs q-ary linear codes from the incidence matrix of the unit graph over certain integer rings and analyzes their duals, proposing conjectures on graph diameter and code properties.

Contribution

It introduces a method to build linear codes from the unit graph of specific rings and studies their duals' minimum distances, also proposing related conjectures.

Findings

Dual codes have minimum distance 3 or 4

Constructed codes are q-ary linear codes from incidence matrices

Proposed conjectures on graph diameter and code properties

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 $n$.