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A $\\mathbb{Z}_\\mu$--{\\it scheme of valency} $(k,l)$ and {\\it order} $(m,n)$ is a $m \\times n$ array $(S_{ij})$ of subsets $S_{ij} \\subseteq \\mathbb{Z}_\\mu$ such that for each row and column one has $\\sum_{j=1}^n |S_{ij}| = k $ and $\\sum_{i=1}^m |S_{ij}| = l$, respectively. Any such scheme is an algebraic equivalent of a $(k,l)$-semi-regular bipartite voltage graph with $n$ and $m$ vertices in the bipartition sets and voltages coming from the cyclic group $\\mathbb{Z}_\\mu$. We are interested in the subclass of $\\mathbb{Z}_\\mu$--schemes that are char"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1111.3265","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-11-14T16:18:34Z","cross_cats_sorted":[],"title_canon_sha256":"cef3e8b841ca88ce020e322b37c4c9b1d2b1c81d0971cd4d13c5ee279cb49815","abstract_canon_sha256":"742e9c518ae708b69a4bcfce86e0e70b6572c8df5224aa59dc4daccb5a711c20"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:29:39.462330Z","signature_b64":"UNS7YzMi91QFbDfvBvRVneN+OI8HQjmqcfTRW0m0nGjx3dEeY6Dj1mvGCNGhE9bPe5JyxIZj+dKUvzg81ImwCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d51586f5b3d94b0020637caca70550caa5f81433ef5a5d5dbf5a14dba171bdc6","last_reissued_at":"2026-05-18T02:29:39.461873Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:29:39.461873Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Ubiquity and Utility of Cyclic Schemes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"D. Labbate, M. Abreu, M. J. Funk, V. Napolitano","submitted_at":"2011-11-14T16:18:34Z","abstract_excerpt":"Let $k,l,m,n$, and $\\mu$ be positive integers. A $\\mathbb{Z}_\\mu$--{\\it scheme of valency} $(k,l)$ and {\\it order} $(m,n)$ is a $m \\times n$ array $(S_{ij})$ of subsets $S_{ij} \\subseteq \\mathbb{Z}_\\mu$ such that for each row and column one has $\\sum_{j=1}^n |S_{ij}| = k $ and $\\sum_{i=1}^m |S_{ij}| = l$, respectively. Any such scheme is an algebraic equivalent of a $(k,l)$-semi-regular bipartite voltage graph with $n$ and $m$ vertices in the bipartition sets and voltages coming from the cyclic group $\\mathbb{Z}_\\mu$. 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