{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:R56X5UQMV6SDNCUOJK7G6POOE3","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"75f7349774c2fe4b4b533aeafa64b97bb003eeab98c6ed6b41d795f4dfaead87","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-01-09T14:56:14Z","title_canon_sha256":"41bb3e58273d0eb6d36b69c530afc486bafc913b28a093beca11ca0c5ae80229"},"schema_version":"1.0","source":{"id":"1401.2022","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1401.2022","created_at":"2026-05-18T03:02:47Z"},{"alias_kind":"arxiv_version","alias_value":"1401.2022v1","created_at":"2026-05-18T03:02:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1401.2022","created_at":"2026-05-18T03:02:47Z"},{"alias_kind":"pith_short_12","alias_value":"R56X5UQMV6SD","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_16","alias_value":"R56X5UQMV6SDNCUO","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_8","alias_value":"R56X5UQM","created_at":"2026-05-18T12:28:46Z"}],"graph_snapshots":[{"event_id":"sha256:fca9002c141560fbd0fc6fc5b38bdfb5a9a096d8dc2aa378495ec596e0f5962a","target":"graph","created_at":"2026-05-18T03:02:47Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"A 3-$(n,4,1)$ packing design consists of an $n$-element set $X$ and a collection of $4$-element subsets of $X$, called {\\it blocks}, such that every $3$-element subset of $X$ is contained in at most one block. The packing number of quadruples $d(3,4,n)$ denotes the number of blocks in a maximum $3$-$(n,4,1)$ packing design, which is also the maximum number $A(n,4,4)$ of codewords in a code of length $n$, constant weight $4$, and minimum Hamming distance 4. In this paper the undecided 21 packing numbers $A(n,4,4)$ are shown to be equal to Johnson bound $J(n,4,4)$ $( =\\lfloor\\frac{n}{4}\\lfloor\\f","authors_text":"Jingjun Bao, Lijun Ji","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-01-09T14:56:14Z","title":"The completion of optimal $(3,4)$-packings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.2022","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:fb04887ef9af2209ab0be0a76c47930cc3f1abfd93b763816f35295280238365","target":"record","created_at":"2026-05-18T03:02:47Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"75f7349774c2fe4b4b533aeafa64b97bb003eeab98c6ed6b41d795f4dfaead87","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-01-09T14:56:14Z","title_canon_sha256":"41bb3e58273d0eb6d36b69c530afc486bafc913b28a093beca11ca0c5ae80229"},"schema_version":"1.0","source":{"id":"1401.2022","kind":"arxiv","version":1}},"canonical_sha256":"8f7d7ed20cafa4368a8e4abe6f3dce26d21a4775967a8de183599b8b7b7f2fa0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8f7d7ed20cafa4368a8e4abe6f3dce26d21a4775967a8de183599b8b7b7f2fa0","first_computed_at":"2026-05-18T03:02:47.748069Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:02:47.748069Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9xIV5TeFMF3nGanSzt47FwWf3VN4DEol8nTvoWV0l0HdU3s0lzefd3jowxbPtXWu8lMiO6I4/FZJWAVIz9lgAg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:02:47.748750Z","signed_message":"canonical_sha256_bytes"},"source_id":"1401.2022","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fb04887ef9af2209ab0be0a76c47930cc3f1abfd93b763816f35295280238365","sha256:fca9002c141560fbd0fc6fc5b38bdfb5a9a096d8dc2aa378495ec596e0f5962a"],"state_sha256":"a5b3effaa23aa657f05fbbf66cd9573dbb2c5c4941757973cc6fe46bf5104d9f"}