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For a nonnegative integer $k$, let $\\mathcal{C}_k$ denote the collection of codes of cardinality at most $k$. For $D \\in \\mathcal{C}_k$, define $S(D) := \\{C \\in \\mathcal{C}_k \\mid D \\subseteq C, |D| +2|C\\setminus D| \\leq k\\}$. Then $N(n_2,n_3,d)$ is upper bounded by the maximum value of $\\sum_{v \\in [2]^{n_2}[3]^{n_3}}x(\\{v\\})$, where $x$ is a function $\\m"},"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":"1606.06930","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-06-22T12:50:08Z","cross_cats_sorted":["math.OC","math.RT"],"title_canon_sha256":"9f9873aca9da0e9214832dd1c1b621401ed3aecf24eb8848323557eed8dc0647","abstract_canon_sha256":"c43de3ed86d033f6dab28a5a16ed7cd400d586e0a7df3fd07c9089a680dfe87c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:19:42.767700Z","signature_b64":"xObLmX+eDYK4QtTyT/tqb8op9ge/4iwnUL2z8BWP8av+e8v2B9BYwQa++0Xju9koNArf+GC6g+Cad3c2bfKZAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"085264880515a3fe39d3a6437a2049bda6dc4fb46652e9f39d9b397e1ebc8659","last_reissued_at":"2026-05-18T00:19:42.767051Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:19:42.767051Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Semidefinite bounds for mixed binary/ternary codes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OC","math.RT"],"primary_cat":"math.CO","authors_text":"Bart Litjens","submitted_at":"2016-06-22T12:50:08Z","abstract_excerpt":"For nonnegative integers $n_2, n_3$ and $d$, let $N(n_2,n_3,d)$ denote the maximum cardinality of a code of length $n_2+n_3$, with $n_2$ binary coordinates and $n_3$ ternary coordinates (in this order) and with minimum distance at least $d$. For a nonnegative integer $k$, let $\\mathcal{C}_k$ denote the collection of codes of cardinality at most $k$. For $D \\in \\mathcal{C}_k$, define $S(D) := \\{C \\in \\mathcal{C}_k \\mid D \\subseteq C, |D| +2|C\\setminus D| \\leq k\\}$. Then $N(n_2,n_3,d)$ is upper bounded by the maximum value of $\\sum_{v \\in [2]^{n_2}[3]^{n_3}}x(\\{v\\})$, where $x$ is a function $\\m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.06930","kind":"arxiv","version":2},"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":"1606.06930","created_at":"2026-05-18T00:19:42.767135+00:00"},{"alias_kind":"arxiv_version","alias_value":"1606.06930v2","created_at":"2026-05-18T00:19:42.767135+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.06930","created_at":"2026-05-18T00:19:42.767135+00:00"},{"alias_kind":"pith_short_12","alias_value":"BBJGJCAFCWR7","created_at":"2026-05-18T12:30:07.202191+00:00"},{"alias_kind":"pith_short_16","alias_value":"BBJGJCAFCWR74OOT","created_at":"2026-05-18T12:30:07.202191+00:00"},{"alias_kind":"pith_short_8","alias_value":"BBJGJCAF","created_at":"2026-05-18T12:30:07.202191+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/BBJGJCAFCWR74OOTUZBXUICJXW","json":"https://pith.science/pith/BBJGJCAFCWR74OOTUZBXUICJXW.json","graph_json":"https://pith.science/api/pith-number/BBJGJCAFCWR74OOTUZBXUICJXW/graph.json","events_json":"https://pith.science/api/pith-number/BBJGJCAFCWR74OOTUZBXUICJXW/events.json","paper":"https://pith.science/paper/BBJGJCAF"},"agent_actions":{"view_html":"https://pith.science/pith/BBJGJCAFCWR74OOTUZBXUICJXW","download_json":"https://pith.science/pith/BBJGJCAFCWR74OOTUZBXUICJXW.json","view_paper":"https://pith.science/paper/BBJGJCAF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1606.06930&json=true","fetch_graph":"https://pith.science/api/pith-number/BBJGJCAFCWR74OOTUZBXUICJXW/graph.json","fetch_events":"https://pith.science/api/pith-number/BBJGJCAFCWR74OOTUZBXUICJXW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BBJGJCAFCWR74OOTUZBXUICJXW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BBJGJCAFCWR74OOTUZBXUICJXW/action/storage_attestation","attest_author":"https://pith.science/pith/BBJGJCAFCWR74OOTUZBXUICJXW/action/author_attestation","sign_citation":"https://pith.science/pith/BBJGJCAFCWR74OOTUZBXUICJXW/action/citation_signature","submit_replication":"https://pith.science/pith/BBJGJCAFCWR74OOTUZBXUICJXW/action/replication_record"}},"created_at":"2026-05-18T00:19:42.767135+00:00","updated_at":"2026-05-18T00:19:42.767135+00:00"}