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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"},"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":"2510.20446","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-10-23T11:30:02Z","cross_cats_sorted":[],"title_canon_sha256":"1d7dd7e353842cc83ed308e7f5f7249a2ae54e70539da5d293e3ab113e465d34","abstract_canon_sha256":"af1f2e12c701015a018ed59d12b568e5508826011060291fd48be72ab8ce28b4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-25T01:17:47.088224Z","signature_b64":"CFeUnwWqXc2vxaWx9YSlBK2Riwq+q2K8qGzjHFVFVRwmpTxAXfc8Xbp5dQI6l6h9r9Po/kBYJgfu9Kt2BqsiBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ab0a8cacb059e43b12cdadcc2e775fb9eaf2b8fd25adf1fc96dd6e7cdf241dd0","last_reissued_at":"2026-06-25T01:17:47.087770Z","signature_status":"signed_v1","first_computed_at":"2026-06-25T01:17:47.087770Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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