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

Let _n[i] be the ring of Gaussian integers modulo a positive integer n. Very recently, Camarero and Martínez [IEEE Trans. Inform. Theory, 62 (2016), 1183–1192], showed that for every prime number p>5 such that p≡± 5 12, the Cayley graph 𝒢_p=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=2⌊p/4⌋. They also conjectured that 𝒢_p is a Ramanujan graph for every prime p such that p≡ 3 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.

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