The Cayley graphs associated with some quasi-perfect Lee codes are Ramanujan graphs

TL;DR

This paper proves that certain Cayley graphs associated with quasi-perfect Lee codes are Ramanujan graphs, confirming a conjecture and connecting spectral graph theory with coding theory.

Contribution

It confirms that Cayley graphs linked to specific quasi-perfect Lee codes are Ramanujan graphs, using Deligne's bounds and eigenvalue analysis, advancing understanding in spectral graph and coding theory.

Findings

Confirmed the conjecture that these Cayley graphs are Ramanujan graphs.

Applied Deligne's bounds to estimate eigenvalues of Cayley graphs.

Provided new insights connecting spectral graph theory, character theory, and coding theory.

Abstract

Let $\Z_n[i]$ be the ring of Gaussian integers modulo a positive integer $n$. Very recently, Camarero and Mart\'{i}nez [IEEE Trans. Inform. Theory, {\bf 62} (2016), 1183--1192], showed that for every prime number $p>5$ such that $p\equiv \pm 5 \pmod{12}$, the Cayley graph $\mathcal{G}_p=\textnormal{Cay}(\Z_p[i], S_2)$, where $S_2$ is the set of units of $\Z_p[i]$, induces a 2-quasi-perfect Lee code over $\Z_p^m$, where $m=2\lfloor \frac{p}{4}\rfloor$. They also conjectured that $\mathcal{G}_p$ is a Ramanujan graph for every prime $p$ such that $p\equiv 3 \pmod{4}$. In this paper, we solve this conjecture. Our main tools are Deligne's bound from 1977 for estimating a particular kind of trigonometric sum and a result of Lov\'{a}sz from 1975 (or of Babai from 1979) which gives the eigenvalues of Cayley graphs of finite Abelian groups. Our proof techniques may motivate more work in the interactions between spectral graph theory, character theory, and coding theory, and may provide new ideas towards the famous Golomb--Welch conjecture on the existence of perfect Lee codes.