Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Cayley graphs associated with some quasi-perfect Lee codes are
  Ramanujan graphs   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.

21
The categories of this paper are: adap-org
8
The categories of this paper are: hep-ex
26
The categories of this paper are: bayes-an
20
The categories of this paper are: acc-phys
15
The categories of this paper are: nucl-ex
28
The categories of this paper are: alg-geom
23
The categories of this paper are: solv-int
29
The categories of this paper are: funct-an
19
The categories of this paper are: eess
13
The categories of this paper are: quant-ph
32
The categories of this paper are: atom-ph
12
The categories of this paper are: nucl-th
7
The categories of this paper are: hep-th
5
The categories of this paper are: gr-qc
18
The categories of this paper are: econ
6
The categories of this paper are: astro-ph