Linear Codes Derived from the Structure of Unit Graphs Over $\mathbb{Z}_n$

TL;DR

The paper analyzes the unit graph G(\mathbb{Z}_n) for n with prime-power factors, proving connectivity and tight diameter bounds, and derives edge-connectivity results. Using incidence matrices of these graphs, it constructs q-ary linear codes and determines their parameters and duals, thereby resolving two conjectures from Jain (2023) on the structural and coding-theoretic properties of unit graphs. The work highlights the interplay between number theory, graph theory, and coding theory, and extends prior results to direct-sum ring decompositions, including cases with a Z_{2^m} component. Overall, it delivers a cohesive framework linking algebraic graph structures to concrete coding-theoretic constructs with explicit parameterizations and dual-distance properties.

Abstract

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.

Theorems & Definitions (18)

• Lemma 1.1
• Theorem 1.2: Chinese Remainder Theorem for $n$-ideals
• Conjecture 2.1
• Conjecture 2.2
• Theorem 2.3
• Theorem 2.4
• proof
• Theorem 2.5
• proof
• Theorem 2.6