{"paper":{"title":"Cyclotomic graphs and perfect codes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Sanming Zhou","submitted_at":"2015-02-11T11:56:46Z","abstract_excerpt":"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 suffi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.03272","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}