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The largest minimum weights $d_2(n,5)$ and $d_3(n,4)$ are partially determined. We also determine the largest minimum weights $d_2(n,n-5)$, $d_3(n,n-i)$ for $i \\in \\{2,3,4\\}$, and $d_3(n,k)$ for $n \\in \\{11,12,\\ldots,19\\}$."},"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":"1908.08661","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.IT","submitted_at":"2019-08-23T04:39:56Z","cross_cats_sorted":["math.CO","math.IT"],"title_canon_sha256":"7c9ddb25aeffad87d0c112da10bb8468ec90247764b3aa9f4f4c32d72c8c2954","abstract_canon_sha256":"17b3def3c20c2ad1c56362393e9d7c7921a15fe78ffeb2583771a11f98db3a96"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:52:33.303501Z","signature_b64":"HWBSH/hkEOa97XBT3f7r+TDf2CNUnTsoIdWAVpMv7+JRdiKsYYvEBsKKmEw9n8ubcxKmpaUIZje2qhLVm+t4Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"45c80c8504ebe75e94b0e769d09a68b8a0f9f42a475e2c1b6d0c019e5640e018","last_reissued_at":"2026-07-05T01:52:33.302955Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:52:33.302955Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the minimum weights of binary LCD codes and ternary LCD codes","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO","math.IT"],"primary_cat":"cs.IT","authors_text":"Ken Saito, Makoto Araya, Masaaki Harada","submitted_at":"2019-08-23T04:39:56Z","abstract_excerpt":"Linear complementary dual (LCD) codes are linear codes that intersect with their dual codes trivially. 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We also determine the largest minimum weights $d_2(n,n-5)$, $d_3(n,n-i)$ for $i \\in \\{2,3,4\\}$, and $d_3(n,k)$ for $n \\in \\{11,12,\\ldots,19\\}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.08661","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.08661/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"1908.08661","created_at":"2026-07-05T01:52:33.303029+00:00"},{"alias_kind":"arxiv_version","alias_value":"1908.08661v3","created_at":"2026-07-05T01:52:33.303029+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.08661","created_at":"2026-07-05T01:52:33.303029+00:00"},{"alias_kind":"pith_short_12","alias_value":"IXEAZBIE5PTV","created_at":"2026-07-05T01:52:33.303029+00:00"},{"alias_kind":"pith_short_16","alias_value":"IXEAZBIE5PTV5FFQ","created_at":"2026-07-05T01:52:33.303029+00:00"},{"alias_kind":"pith_short_8","alias_value":"IXEAZBIE","created_at":"2026-07-05T01:52:33.303029+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/IXEAZBIE5PTV5FFQ45U5BGTIXC","json":"https://pith.science/pith/IXEAZBIE5PTV5FFQ45U5BGTIXC.json","graph_json":"https://pith.science/api/pith-number/IXEAZBIE5PTV5FFQ45U5BGTIXC/graph.json","events_json":"https://pith.science/api/pith-number/IXEAZBIE5PTV5FFQ45U5BGTIXC/events.json","paper":"https://pith.science/paper/IXEAZBIE"},"agent_actions":{"view_html":"https://pith.science/pith/IXEAZBIE5PTV5FFQ45U5BGTIXC","download_json":"https://pith.science/pith/IXEAZBIE5PTV5FFQ45U5BGTIXC.json","view_paper":"https://pith.science/paper/IXEAZBIE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1908.08661&json=true","fetch_graph":"https://pith.science/api/pith-number/IXEAZBIE5PTV5FFQ45U5BGTIXC/graph.json","fetch_events":"https://pith.science/api/pith-number/IXEAZBIE5PTV5FFQ45U5BGTIXC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IXEAZBIE5PTV5FFQ45U5BGTIXC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IXEAZBIE5PTV5FFQ45U5BGTIXC/action/storage_attestation","attest_author":"https://pith.science/pith/IXEAZBIE5PTV5FFQ45U5BGTIXC/action/author_attestation","sign_citation":"https://pith.science/pith/IXEAZBIE5PTV5FFQ45U5BGTIXC/action/citation_signature","submit_replication":"https://pith.science/pith/IXEAZBIE5PTV5FFQ45U5BGTIXC/action/replication_record"}},"created_at":"2026-07-05T01:52:33.303029+00:00","updated_at":"2026-07-05T01:52:33.303029+00:00"}