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 $q = p^k \ge 14$, $p\geq 5$ a prime. Our codes are derived from the generating set $H_q = \{(a, a^3) \mid a \in \mathbb{F}_q^*\}$ 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.

翻译：本文针对素数$p\geq 5$及$q = p^k \ge 14$，提出了一类新的显式无限族$2$-准完美$p$元Lee码，其长度为$\frac{q-1}{2}$，维数为$\frac{q-1}{2}-2k$。我们的码系由有限域$\mathbb{F}_{q^2}$加法群的生成集$H_q = \{(a, a^3) \mid a \in \mathbb{F}_q^*\}$导出。此外，我们建立了由Mesnager、Tang和Qi构造的$2$-准完美Lee码与著名的阿贝尔Ramanujan图（即Li图与有限欧几里得图）之间的桥梁，为这些族提供了统一的理论框架。