$2$-quasi-perfect Lee codes and abelian Ramanujan graphs: a new construction and relationship
For coding theorists, this provides new explicit codes with good parameters and reveals a structural link to Ramanujan graphs, though the construction is incremental over existing methods.
This paper constructs an explicit infinite family of 2-quasi-perfect p-ary Lee codes with length (q-1)/2 and dimension (q-1)/2 - 2k for q = p^k ≥ 14, p ≥ 5 prime, and establishes a connection between these codes and abelian Ramanujan graphs, providing a unified theoretical framework.
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 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.