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Hanani \\cite{Hanani} proved that the necessary condition $n\\ {\\rm mod}\\ 6= 2\\ {\\rm or}\\ 4$ for the existence of a Steiner quadruple systems of order $n$ is also sufficient. Lenz \\cite{Lenz} proved that the logarithm of the number of different $SQS(n)$ is greater than $cn^3$ where $c>0$ is a constant and $n$ is admissible. We prove that the logarithm of the number of different $SQS(n)$ is $\\Theta(n^3\\"},"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":"1606.02426","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-06-08T07:10:00Z","cross_cats_sorted":[],"title_canon_sha256":"392d213ffa5d4deb1f8c89669d765face7609778e1ee9bed3c03d273b19cc885","abstract_canon_sha256":"e4b09aec2feb85cc05b7891803ab010334c06f7264a3ec0bd748a624df5f15da"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:38:42.261443Z","signature_b64":"14e23rG/zLoavRVg6H2yDKCSCCImke89Etq3WEJXbqqfoMqSAzRBlGsZqpIsKE1F48wMJpblwBrwDRbbGeZLDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"695ceeae2be0fe57049c76beb4c0d69eb12ffbb1f08335c11a2b55ab2cd9f3b2","last_reissued_at":"2026-05-18T00:38:42.260597Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:38:42.260597Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the number of SQS","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Vladimir N. Potapov","submitted_at":"2016-06-08T07:10:00Z","abstract_excerpt":"A Steiner quadruple system (briefly $SQS(n)$) is a pair $(X,B)$ where $|X|=n$ and $B$ is a collection of 4-element blocks such that every 3-subset of $X$ is contained in exactly one member of $B$. Hanani \\cite{Hanani} proved that the necessary condition $n\\ {\\rm mod}\\ 6= 2\\ {\\rm or}\\ 4$ for the existence of a Steiner quadruple systems of order $n$ is also sufficient. Lenz \\cite{Lenz} proved that the logarithm of the number of different $SQS(n)$ is greater than $cn^3$ where $c>0$ is a constant and $n$ is admissible. We prove that the logarithm of the number of different $SQS(n)$ is $\\Theta(n^3\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.02426","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":""},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1606.02426","created_at":"2026-05-18T00:38:42.260737+00:00"},{"alias_kind":"arxiv_version","alias_value":"1606.02426v1","created_at":"2026-05-18T00:38:42.260737+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.02426","created_at":"2026-05-18T00:38:42.260737+00:00"},{"alias_kind":"pith_short_12","alias_value":"NFOO5LRL4D7F","created_at":"2026-05-18T12:30:32.724797+00:00"},{"alias_kind":"pith_short_16","alias_value":"NFOO5LRL4D7FOBE4","created_at":"2026-05-18T12:30:32.724797+00:00"},{"alias_kind":"pith_short_8","alias_value":"NFOO5LRL","created_at":"2026-05-18T12:30:32.724797+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/NFOO5LRL4D7FOBE4O27LJQGWT2","json":"https://pith.science/pith/NFOO5LRL4D7FOBE4O27LJQGWT2.json","graph_json":"https://pith.science/api/pith-number/NFOO5LRL4D7FOBE4O27LJQGWT2/graph.json","events_json":"https://pith.science/api/pith-number/NFOO5LRL4D7FOBE4O27LJQGWT2/events.json","paper":"https://pith.science/paper/NFOO5LRL"},"agent_actions":{"view_html":"https://pith.science/pith/NFOO5LRL4D7FOBE4O27LJQGWT2","download_json":"https://pith.science/pith/NFOO5LRL4D7FOBE4O27LJQGWT2.json","view_paper":"https://pith.science/paper/NFOO5LRL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1606.02426&json=true","fetch_graph":"https://pith.science/api/pith-number/NFOO5LRL4D7FOBE4O27LJQGWT2/graph.json","fetch_events":"https://pith.science/api/pith-number/NFOO5LRL4D7FOBE4O27LJQGWT2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NFOO5LRL4D7FOBE4O27LJQGWT2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NFOO5LRL4D7FOBE4O27LJQGWT2/action/storage_attestation","attest_author":"https://pith.science/pith/NFOO5LRL4D7FOBE4O27LJQGWT2/action/author_attestation","sign_citation":"https://pith.science/pith/NFOO5LRL4D7FOBE4O27LJQGWT2/action/citation_signature","submit_replication":"https://pith.science/pith/NFOO5LRL4D7FOBE4O27LJQGWT2/action/replication_record"}},"created_at":"2026-05-18T00:38:42.260737+00:00","updated_at":"2026-05-18T00:38:42.260737+00:00"}