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The Cayley graphs associated with some quasi-perfect Lee codes are Ramanujan graphs | Khodakhast Bibak
; Bruce M. Kapron
; Venkatesh Srinivasan
; | Date: |
12 Oct 2020 | Abstract: | Let $_n[i]$ be the ring of Gaussian integers modulo a positive integer $n$.
Very recently, Camarero and Martínez [IEEE Trans. Inform. Theory, {f 62}
(2016), 1183--1192], showed that for every prime number $p>5$ such that
$pequiv pm 5 pmod{12}$, the Cayley graph
$mathcal{G}_p= extnormal{Cay}(_p[i], S_2)$, where $S_2$ is the set of units
of $_p[i]$, induces a 2-quasi-perfect Lee code over $_p^m$, where
$m=2lfloor frac{p}{4}
floor$. They also conjectured that $mathcal{G}_p$ is
a Ramanujan graph for every prime $p$ such that $pequiv 3 pmod{4}$. 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
Lovász 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. | Source: | arXiv, 2010.05407 | Services: | Forum | Review | PDF | Favorites |
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