{"paper":{"title":"Perfect difference families, perfect systems of difference sets and their applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hengrui Liu, Menglong Zhang, Tao Feng, Xiaomiao Wang","submitted_at":"2025-10-23T11:30:02Z","abstract_excerpt":"Let $v$ be a positive odd integer. A $(v,k,\\lambda)$-perfect difference family (PDF) is a collection $\\mathcal{F}$ of $k$-subsets of $\\{0,1,\\ldots,v-1\\}$ such that the multiset $\\bigcup_{F\\in \\mathcal{F}}\\{x-y : x,y\\in F, x>y\\}$ covers each element of $\\left\\{1,2,\\ldots,(v-1)/2\\right\\}$ exactly $\\lambda$ times. Perfect difference families are a special class of perfect systems of difference sets. They were introduced by Bermond, Kotzig, and Turgeon in the 1970s, following a problem suggested by Erd\\H{o}s. In this paper, we prove that a $(v,4,\\lambda)$-PDF exists if and only if $\\lambda(v-1) \\e"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2510.20446","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2510.20446/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}