{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:QIIYQYNWS22QCYMRI2LJY4HD2G","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":"87a8359e8fa0e67dd3b5c21b23fb3ec2d9efce5eb706bcc8b280e7509bc54f71","cross_cats_sorted":["cs.IT","math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2015-11-28T10:49:41Z","title_canon_sha256":"215ca62889a5f5e8f18891b3d0bf08dfd561cbaebdd18ed9548be134ad20b5a1"},"schema_version":"1.0","source":{"id":"1511.08889","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.08889","created_at":"2026-05-18T01:25:46Z"},{"alias_kind":"arxiv_version","alias_value":"1511.08889v1","created_at":"2026-05-18T01:25:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.08889","created_at":"2026-05-18T01:25:46Z"},{"alias_kind":"pith_short_12","alias_value":"QIIYQYNWS22Q","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_16","alias_value":"QIIYQYNWS22QCYMR","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_8","alias_value":"QIIYQYNW","created_at":"2026-05-18T12:29:37Z"}],"graph_snapshots":[{"event_id":"sha256:a80b028da27a50b614cf6879415c6361abae8f059c456e61270d26ee038c4dda","target":"graph","created_at":"2026-05-18T01:25:46Z","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 binary linear code is called {\\em LCD} if it intersects its dual trivially. We show that the coefficients of the joint weight enumerator of such a code with its dual satisfy linear constraints, leading to a new linear programming bound on the size of an LCD code of given length and minimum distance. In addition, we show that this polynomial is, in general, an invariant of a matrix group of dimension $4$ and order $12$. Also, we sketch a Gleason formula for this weight enumerator.","authors_text":"Adel Alahmadi, Mathieu Dutour Sikiri\\'c, Michel Deza, Patrick Sol\\'e","cross_cats":["cs.IT","math.IT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2015-11-28T10:49:41Z","title":"The joint weight enumerator of an LCD code and its dual"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.08889","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":"sha256:0c7ba9df51614a9c790917b3d880e426136f0e3abfa31757d4b57f3031ed3790","target":"record","created_at":"2026-05-18T01:25:46Z","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":"87a8359e8fa0e67dd3b5c21b23fb3ec2d9efce5eb706bcc8b280e7509bc54f71","cross_cats_sorted":["cs.IT","math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2015-11-28T10:49:41Z","title_canon_sha256":"215ca62889a5f5e8f18891b3d0bf08dfd561cbaebdd18ed9548be134ad20b5a1"},"schema_version":"1.0","source":{"id":"1511.08889","kind":"arxiv","version":1}},"canonical_sha256":"82118861b696b501619146969c70e3d1997fb9d34159e364cd944885acc95d68","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"82118861b696b501619146969c70e3d1997fb9d34159e364cd944885acc95d68","first_computed_at":"2026-05-18T01:25:46.165718Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:25:46.165718Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"OtOkLry/Aa7QJ9YJAo2MszMsPN5NmN+9C0Cn2DRb6NMI6KKxQIsZDv52Kq/VV338BVEx71tozIXs+3n9pHnTAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:25:46.166206Z","signed_message":"canonical_sha256_bytes"},"source_id":"1511.08889","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0c7ba9df51614a9c790917b3d880e426136f0e3abfa31757d4b57f3031ed3790","sha256:a80b028da27a50b614cf6879415c6361abae8f059c456e61270d26ee038c4dda"],"state_sha256":"76592965d318ee4229d7e1176b0365485ba4467bd690a5748842ed3cbd86cc3a"}