{"paper":{"title":"Necessary Conditions for the Existence of Group-Invariant Butson Matrices and a New Family of Perfect Arrays","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Tai Do Duc","submitted_at":"2019-03-18T09:52:19Z","abstract_excerpt":"Let $G$ be a finite abelian group and let $\\exp(G)$ denote the least common multiple of the orders of all elements of $G$. A $BH(G,h)$ matrix is a $G$-invariant $|G|\\times |G|$ matrix $H$ whose entries are complex $h$th roots of unity such that $HH^*=|G|I_{|G|}$. In this paper, we study the relation between $G$ and $h$ so that a $BH(G,h)$ matrix exists. We will only focus on $BH(\\mathbb{Z}_n,h)$ matrices and $BH(G,2p^b)$ matrices, where $p$ is an odd prime. By our results, there are $2687$ open cases left for the existence of $BH(\\mathbb{Z}_n,h)$ matrices in which $1\\leq n,h \\leq 100$. In the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.07334","kind":"arxiv","version":2},"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"}