论文标题
与某些准完美的Lee代码相关的Cayley图是Ramanujan图
The Cayley graphs associated with some quasi-perfect Lee codes are Ramanujan graphs
论文作者
论文摘要
令$ \ z_n [i] $为高斯整数的戒指，一个正整数$ n $。最近，Camarero和Martínez[IEEE Trans。通知。理论，{\ bf 62}（2016），1183--1192]表明，对于每个素数$ p> 5 $，以至于$ p \ equiv \ equiv \ pm 5 \ pmod {12} $，cayley graph $ \ mathcal $ \ nathcal {g} _p = \ textnormal $ \ z_p [i] $的单位，在$ \ z_p^m $上诱导2 quasi Perfect Lee代码，其中$ m = 2 \ lfloor \ lfloor \ frac {p} {4} {4} \ rfloor $。他们还猜想$ \ Mathcal {g} _p $是每个Prime $ p $的Ramanujan图，因此$ p \ equiv 3 \ pmod {4} $。在本文中，我们解决了这个猜想。我们的主要工具是从1977年开始的Deligne's Bound，用于估计一种特殊的三角总和，以及1975年的Lovász（或1979年的Babai）的结果，这给出了有限阿伯利亚群体Cayley图的特征值。我们的证明技术可能会激励在频谱图理论，性格理论和编码理论之间的相互作用中进行更多的工作，并可能为著名的Golomb提供新的思想 - 对完美Lee代码的存在。
Let $\Z_n[i]$ be the ring of Gaussian integers modulo a positive integer $n$. Very recently, Camarero and Martínez [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 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.
