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

This paper presents a new explicit infinite family of 2-quasi-perfect $p$-ary Lee codes of length $\frac{q-1}{2}$ and dimension $\frac{q-1}{2}-2k$ for $q=p^k\ge 14$, $p\ge 5$ a prime. Our codes are derived from the generating set $H_q=\{(a,a^3)\mid a\in {\mathbb{F}}_q^{\ast }\}$ of the additive group of the finite field ${\mathbb{F}}_{q^2}$. Furthermore, we bridge between 2-quasi-perfect Lee codes constructed by Mesnager, Tang, and Qi and well-known abelian Ramanujan graphs, specifically Li’s graphs and finite Euclidean graphs, providing a unified theoretical framework for these families.