{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:Y646RWV64EDCEGKVL5J47WJCDF","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":"298169fd401ef9bde72e301351aba4b7fb5771f11e4ada6fd426b4e8b6bfbe0c","cross_cats_sorted":["math.IT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.IT","submitted_at":"2025-06-09T03:02:00Z","title_canon_sha256":"71704b5e85e4e95441cee07087aaef6f7809553d6cca41b52ceedcd556033a94"},"schema_version":"1.0","source":{"id":"2506.07380","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.07380","created_at":"2026-07-05T11:18:26Z"},{"alias_kind":"arxiv_version","alias_value":"2506.07380v1","created_at":"2026-07-05T11:18:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.07380","created_at":"2026-07-05T11:18:26Z"},{"alias_kind":"pith_short_12","alias_value":"Y646RWV64EDC","created_at":"2026-07-05T11:18:26Z"},{"alias_kind":"pith_short_16","alias_value":"Y646RWV64EDCEGKV","created_at":"2026-07-05T11:18:26Z"},{"alias_kind":"pith_short_8","alias_value":"Y646RWV6","created_at":"2026-07-05T11:18:26Z"}],"graph_snapshots":[{"event_id":"sha256:30c962539921a91a95e5401492bce8ac9112000f316ba0bd72ca23b071099c5d","target":"graph","created_at":"2026-07-05T11:18:26Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2506.07380/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The error-correcting pair is a general algebraic decoding method for linear codes. The near maximal distance separable (NMDS) linear code is a subclass of linear codes and has applications in secret sharing scheme and communication systems due to the efficient performance, thus we focus on the error-correcting pair of NMDS linear codes. In 2023, He and Liao showed that for an NMDS linear code $\\mathcal{C}$ with minimal distance $2\\ell+1$ or $2\\ell+2$, if $\\mathcal{C}$ has an $\\ell$-error-correcting pair $\\left( \\mathcal{A}, \\mathcal{B} \\right)$, then the parameters of $\\mathcal{A}$ have 6 or 1","authors_text":"Dong He, Qunying Liao, Zhaohui Zhang","cross_cats":["math.IT"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.IT","submitted_at":"2025-06-09T03:02:00Z","title":"The error-correcting pair for several classes of NMDS linear codes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.07380","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:ba6ccf275361a8e9cd7e36a7fd39fac2d617bb1935ee9ade7dec96be8fcb56e3","target":"record","created_at":"2026-07-05T11:18:26Z","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":"298169fd401ef9bde72e301351aba4b7fb5771f11e4ada6fd426b4e8b6bfbe0c","cross_cats_sorted":["math.IT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.IT","submitted_at":"2025-06-09T03:02:00Z","title_canon_sha256":"71704b5e85e4e95441cee07087aaef6f7809553d6cca41b52ceedcd556033a94"},"schema_version":"1.0","source":{"id":"2506.07380","kind":"arxiv","version":1}},"canonical_sha256":"c7b9e8dabee1062219555f53cfd922197c0c69c3919557ee2425f92fe5d5bc08","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c7b9e8dabee1062219555f53cfd922197c0c69c3919557ee2425f92fe5d5bc08","first_computed_at":"2026-07-05T11:18:26.205119Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:18:26.205119Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"lA0wJd+yXyrbzGnuXipFySVpxMYsl2gRfjRMtJurJfsnVRDmhAgzbu4GScRO3QJwjiGwyr7/A2mxp1N9jDy6Cw==","signature_status":"signed_v1","signed_at":"2026-07-05T11:18:26.205777Z","signed_message":"canonical_sha256_bytes"},"source_id":"2506.07380","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ba6ccf275361a8e9cd7e36a7fd39fac2d617bb1935ee9ade7dec96be8fcb56e3","sha256:30c962539921a91a95e5401492bce8ac9112000f316ba0bd72ca23b071099c5d"],"state_sha256":"360fede05096ff248188fe672a8c3c1bcfb7bc6991ebb91db4361e22a4d5e114"}