$2$-quasi-perfect Lee codes and abelian Ramanujan graphs: a new construction and relationship

TL;DR

This work advances $2$-quasi-perfect Lee codes by providing a new explicit construction using $H_q=\{(a,a^3)\mid a\in \mathbb{F}_q^*\}$, producing codes of length $\\frac{q-1}{2}$ for $q=p^k\ge 14$ with dimension $\\frac{q-1}{2}-2k$, and shows the associated Cayley graphs are almost Ramanujan. It also strengthens the bridge between coding theory and spectral graph theory by connecting the Mesnager–Tang–Qi codes to abelian Ramanujan graphs (Li's graphs and finite Euclidean graphs) and by situating the new construction within the broader MTQ framework. The results leverage Weil-sum estimates to establish almost-Ramanujan properties and rely on a Ramanujan-based diameter rationale to achieve $2$-quasi-perfection. Overall, the paper broadens explicit quasi-perfect Lee-code constructions and highlights deep connections between Lee codes, abelian Ramanujan graphs, and finite-geometry techniques.

Abstract

In this paper, we obtain a new explicit family of $2$-quasi-perfect Lee codes of arbitrarily large length. Our construction is based on generating sets of abelian (almost) Ramanujan graphs obtained by Forey, Fresán, Kowalski and Wigderson. Also, we develop a relationship between certain abelian Ramanujan graphs and $2$-quasi-perfect Lee codes obtained by Mesnager, Tang and Qi.

Theorems & Definitions (27)

• Definition 1: MTQ2018
• Lemma 2: Proposition 6 in MTQ2018
• Remark 3
• Theorem 4
• Remark 5
• Remark 6
• Example 7
• Example 8
• Example 9
• Example 10