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 {±(ζm i+A):0≤i≤m−1} and {±(ζm i+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 kn..