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Here we show uniqueness of the codes achieving these bounds.\n  Let $A(n,d)$ denote the maximum size of a binary code of word length $n$ and minimum distance $d$. Gijswijt, Mittelmann and Schrijver showed that $A(20,8)=256$. We show that there are several nonisomorphic codes achieving this bound, and cl"},"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":"1709.02195","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-09-07T12:09:04Z","cross_cats_sorted":["math.OC"],"title_canon_sha256":"93e1ef46271bf4b42fe81773d73c2af08d55b7d9a04e6aa46a542e8e81f11fee","abstract_canon_sha256":"742f29487fffcce1e92becc58ada63127798a465ea7e94ac75748bc28ceea4aa"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:59:33.183738Z","signature_b64":"rQkqxelU8bive2lFSrIgl6Pvhw56cXoKua+G1eImZe3AzuoszfE9ptj1Lzhu/ll8AhBfGc4A9aDAFTcqQTMpAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"51d6bdb7cea3d9813fe910757205ef819ad81fd56d6e76a23ae87e51a2d71bd6","last_reissued_at":"2026-05-17T23:59:33.183143Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:59:33.183143Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Uniqueness of codes using semidefinite programming","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OC"],"primary_cat":"math.CO","authors_text":"Andries E. Brouwer, Sven C. Polak","submitted_at":"2017-09-07T12:09:04Z","abstract_excerpt":"For $n,d,w \\in \\mathbb{N}$, let $A(n,d,w)$ denote the maximum size of a binary code of word length $n$, minimum distance $d$ and constant weight $w$. Schrijver recently showed using semidefinite programming that $A(23,8,11)=1288$, and the second author that $A(22,8,11)=672$ and $A(22,8,10)=616$. Here we show uniqueness of the codes achieving these bounds.\n  Let $A(n,d)$ denote the maximum size of a binary code of word length $n$ and minimum distance $d$. Gijswijt, Mittelmann and Schrijver showed that $A(20,8)=256$. We show that there are several nonisomorphic codes achieving this bound, and cl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.02195","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":"1709.02195","created_at":"2026-05-17T23:59:33.183259+00:00"},{"alias_kind":"arxiv_version","alias_value":"1709.02195v2","created_at":"2026-05-17T23:59:33.183259+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.02195","created_at":"2026-05-17T23:59:33.183259+00:00"},{"alias_kind":"pith_short_12","alias_value":"KHLL3N6OUPMY","created_at":"2026-05-18T12:31:24.725408+00:00"},{"alias_kind":"pith_short_16","alias_value":"KHLL3N6OUPMYCP7J","created_at":"2026-05-18T12:31:24.725408+00:00"},{"alias_kind":"pith_short_8","alias_value":"KHLL3N6O","created_at":"2026-05-18T12:31:24.725408+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/KHLL3N6OUPMYCP7JCB2XEBPPQG","json":"https://pith.science/pith/KHLL3N6OUPMYCP7JCB2XEBPPQG.json","graph_json":"https://pith.science/api/pith-number/KHLL3N6OUPMYCP7JCB2XEBPPQG/graph.json","events_json":"https://pith.science/api/pith-number/KHLL3N6OUPMYCP7JCB2XEBPPQG/events.json","paper":"https://pith.science/paper/KHLL3N6O"},"agent_actions":{"view_html":"https://pith.science/pith/KHLL3N6OUPMYCP7JCB2XEBPPQG","download_json":"https://pith.science/pith/KHLL3N6OUPMYCP7JCB2XEBPPQG.json","view_paper":"https://pith.science/paper/KHLL3N6O","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1709.02195&json=true","fetch_graph":"https://pith.science/api/pith-number/KHLL3N6OUPMYCP7JCB2XEBPPQG/graph.json","fetch_events":"https://pith.science/api/pith-number/KHLL3N6OUPMYCP7JCB2XEBPPQG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KHLL3N6OUPMYCP7JCB2XEBPPQG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KHLL3N6OUPMYCP7JCB2XEBPPQG/action/storage_attestation","attest_author":"https://pith.science/pith/KHLL3N6OUPMYCP7JCB2XEBPPQG/action/author_attestation","sign_citation":"https://pith.science/pith/KHLL3N6OUPMYCP7JCB2XEBPPQG/action/citation_signature","submit_replication":"https://pith.science/pith/KHLL3N6OUPMYCP7JCB2XEBPPQG/action/replication_record"}},"created_at":"2026-05-17T23:59:33.183259+00:00","updated_at":"2026-05-17T23:59:33.183259+00:00"}