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

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.