Cyclotomic graphs and perfect codes

TL;DR

This paper investigates cyclotomic graphs and perfect codes within them, providing necessary and sufficient conditions for perfect codes, classifying certain Frobenius circulants, and exploring their applications in efficient network routing.

Contribution

It introduces new conditions for perfect codes in cyclotomic graphs, classifies first kind Frobenius circulants as cyclotomic graphs, and links these graphs to efficient routing networks.

Findings

Necessary and sufficient conditions for perfect codes in cyclotomic graphs.

Classification of first kind Frobenius circulants as pth cyclotomic graphs.

Identification of these graphs as efficient for routing and gossiping.

Abstract

We study two families of cyclotomic graphs and perfect codes in them. They are Cayley graphs on the additive group of $\mathbb{Z}[\zeta_m]/A$, with connection sets $\{\pm (\zeta_m^i + A): 0 \le i \le m-1\}$ and $\{\pm (\zeta_m^i + A): 0 \le i \le \phi(m) - 1\}$, respectively, where $\zeta_m$ ($m \ge 2$) is an $m$th primitive root of unity, $A$ a nonzero ideal of $\mathbb{Z}[\zeta_m]$, and $\phi$ Euler's totient function. We call them the $m$th cyclotomic graph and the second kind $m$th cyclotomic graph, and denote them by $G_{m}(A)$ and $G^*_{m}(A)$, respectively. We give a necessary and sufficient condition for $D/A$ to be a perfect $t$-code in $G^*_{m}(A)$ and a necessary condition for $D/A$ to be such a code in $G_{m}(A)$, where $t \ge 1$ is an integer and $D$ an ideal of $\mathbb{Z}[\zeta_m]$ containing $A$. In the case when $m = 3, 4$, $G_m((\alpha))$ is known as an Eisenstein-Jacobi and Gaussian networks, respectively, and we obtain necessary conditions for $(\beta)/(\alpha)$ to be a perfect $t$-code in $G_m((\alpha))$, where $0 \ne \alpha, \beta \in \mathbb{Z}[\zeta_m]$ with $\beta$ dividing $\alpha$. In the literature such conditions are known to be sufficient when $m=4$ and $m=3$ under an additional condition. We give a classification of all first kind Frobenius circulants of valency $2p$ and prove that they are all $p$th cyclotomic graphs, where $p$ is an odd prime. Such graphs belong to a large family of Cayley graphs that are efficient for routing and gossiping.