Cyclotomic graphs and perfect codes

Abstract

We study two families of cyclotomic graphs and perfect codes in them. They are Cayley graphs on the additive group of Z[ζm]/A, with connection sets {±(ζmi+A):0≤i≤m−1} and {±(ζmi+A):0≤i≤ϕ(m)−1}, respectively, where ζm (m≥2) is an mth primitive root of unity, A a nonzero ideal of Z[ζm], and ϕ Euler's totient function. We call them the mth cyclotomic graph and the second kind mth cyclotomic graph, and denote them by Gm(A) and Gm⁎(A), respectively. We give a necessary and sufficient condition for D/A to be a perfect t-code in Gm⁎(A) and a necessary condition for D/A to be such a code in Gm(A), where t≥1 is an integer and D an ideal of Z[ζm] containing A. In the case when m=3,4, Gm((α)) is known as an Eisenstein–Jacobi and Gaussian networks, respectively, and we obtain necessary conditions for (β)/(α) to be a perfect t-code in Gm((α)), where 0≠α,β∈Z[ζm] with β dividing α. 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 pth 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.