{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:2N24CP6GLHLWB6QM54PFNFVQ5Y","short_pith_number":"pith:2N24CP6G","schema_version":"1.0","canonical_sha256":"d375c13fc659d760fa0cef1e5696b0ee0892f0cb8a8b7a8b54ea92129453a2cd","source":{"kind":"arxiv","id":"1606.05144","version":2},"attestation_state":"computed","paper":{"title":"New nonbinary code bounds based on divisibility arguments","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Sven Polak","submitted_at":"2016-06-16T11:17:09Z","abstract_excerpt":"For $q,n,d \\in \\mathbb{N}$, let $A_q(n,d)$ be the maximum size of a code $C \\subseteq [q]^n$ with minimum distance at least $d$. We give a divisibility argument resulting in the new upper bounds $A_5(8,6) \\leq 65$, $A_4(11,8)\\leq 60$ and $A_3(16,11) \\leq 29$. These in turn imply the new upper bounds $A_5(9,6) \\leq 325$, $A_5(10,6) \\leq 1625$, $A_5(11,6) \\leq 8125$ and $A_4(12,8) \\leq 240$. Furthermore, we prove that for $\\mu,q \\in \\mathbb{N}$, there is a 1-1-correspondence between symmetric $(\\mu,q)$-nets (which are certain designs) and codes $C \\subseteq [q]^{\\mu q}$ of size $\\mu q^2$ with mi"},"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":"1606.05144","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-06-16T11:17:09Z","cross_cats_sorted":[],"title_canon_sha256":"58cc8cdc0c99117876944862f98e820de8030fea373f202510e34e12a2382b19","abstract_canon_sha256":"713ef93ee44d7c940d4f42a1cfbaac9f069cea59285dd7069bb6f9863e56dd23"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:08:57.924752Z","signature_b64":"j2yiPsoPJ8vPj+AldD2rMT0kMu681SIf21jyEi4+UfjIU5iYDrPbH/0yRDbB6joDYBKEOkbPUgAgqFg78caHDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d375c13fc659d760fa0cef1e5696b0ee0892f0cb8a8b7a8b54ea92129453a2cd","last_reissued_at":"2026-05-18T00:08:57.923912Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:08:57.923912Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"New nonbinary code bounds based on divisibility arguments","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Sven Polak","submitted_at":"2016-06-16T11:17:09Z","abstract_excerpt":"For $q,n,d \\in \\mathbb{N}$, let $A_q(n,d)$ be the maximum size of a code $C \\subseteq [q]^n$ with minimum distance at least $d$. We give a divisibility argument resulting in the new upper bounds $A_5(8,6) \\leq 65$, $A_4(11,8)\\leq 60$ and $A_3(16,11) \\leq 29$. These in turn imply the new upper bounds $A_5(9,6) \\leq 325$, $A_5(10,6) \\leq 1625$, $A_5(11,6) \\leq 8125$ and $A_4(12,8) \\leq 240$. Furthermore, we prove that for $\\mu,q \\in \\mathbb{N}$, there is a 1-1-correspondence between symmetric $(\\mu,q)$-nets (which are certain designs) and codes $C \\subseteq [q]^{\\mu q}$ of size $\\mu q^2$ with mi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.05144","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":"1606.05144","created_at":"2026-05-18T00:08:57.924044+00:00"},{"alias_kind":"arxiv_version","alias_value":"1606.05144v2","created_at":"2026-05-18T00:08:57.924044+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.05144","created_at":"2026-05-18T00:08:57.924044+00:00"},{"alias_kind":"pith_short_12","alias_value":"2N24CP6GLHLW","created_at":"2026-05-18T12:29:55.572404+00:00"},{"alias_kind":"pith_short_16","alias_value":"2N24CP6GLHLWB6QM","created_at":"2026-05-18T12:29:55.572404+00:00"},{"alias_kind":"pith_short_8","alias_value":"2N24CP6G","created_at":"2026-05-18T12:29:55.572404+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/2N24CP6GLHLWB6QM54PFNFVQ5Y","json":"https://pith.science/pith/2N24CP6GLHLWB6QM54PFNFVQ5Y.json","graph_json":"https://pith.science/api/pith-number/2N24CP6GLHLWB6QM54PFNFVQ5Y/graph.json","events_json":"https://pith.science/api/pith-number/2N24CP6GLHLWB6QM54PFNFVQ5Y/events.json","paper":"https://pith.science/paper/2N24CP6G"},"agent_actions":{"view_html":"https://pith.science/pith/2N24CP6GLHLWB6QM54PFNFVQ5Y","download_json":"https://pith.science/pith/2N24CP6GLHLWB6QM54PFNFVQ5Y.json","view_paper":"https://pith.science/paper/2N24CP6G","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1606.05144&json=true","fetch_graph":"https://pith.science/api/pith-number/2N24CP6GLHLWB6QM54PFNFVQ5Y/graph.json","fetch_events":"https://pith.science/api/pith-number/2N24CP6GLHLWB6QM54PFNFVQ5Y/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2N24CP6GLHLWB6QM54PFNFVQ5Y/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2N24CP6GLHLWB6QM54PFNFVQ5Y/action/storage_attestation","attest_author":"https://pith.science/pith/2N24CP6GLHLWB6QM54PFNFVQ5Y/action/author_attestation","sign_citation":"https://pith.science/pith/2N24CP6GLHLWB6QM54PFNFVQ5Y/action/citation_signature","submit_replication":"https://pith.science/pith/2N24CP6GLHLWB6QM54PFNFVQ5Y/action/replication_record"}},"created_at":"2026-05-18T00:08:57.924044+00:00","updated_at":"2026-05-18T00:08:57.924044+00:00"}