{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:S3JCHREY7TIALNJTA4MVQZNYQW","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":"9c941f3d703f9dc5195411769994531c10f4a7f9bc36370874ab0b13180f74a1","cross_cats_sorted":["math.IT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.IT","submitted_at":"2026-01-28T13:38:17Z","title_canon_sha256":"537fff4aac1b51b65a0f094614a8a4f30b18ff02ca89b7bc6b84877b3286732b"},"schema_version":"1.0","source":{"id":"2601.20600","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2601.20600","created_at":"2026-06-09T02:07:18Z"},{"alias_kind":"arxiv_version","alias_value":"2601.20600v2","created_at":"2026-06-09T02:07:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2601.20600","created_at":"2026-06-09T02:07:18Z"},{"alias_kind":"pith_short_12","alias_value":"S3JCHREY7TIA","created_at":"2026-06-09T02:07:18Z"},{"alias_kind":"pith_short_16","alias_value":"S3JCHREY7TIALNJT","created_at":"2026-06-09T02:07:18Z"},{"alias_kind":"pith_short_8","alias_value":"S3JCHREY","created_at":"2026-06-09T02:07:18Z"}],"graph_snapshots":[{"event_id":"sha256:b289769c431c274039e8d32291ccc0df3ab4f0f169591d3ecb6b468e22c0265a","target":"graph","created_at":"2026-06-09T02:07:18Z","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/2601.20600/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In the recent years, there has been active research on self-orthogonal embeddings of linear codes since they yielded some optimal self-orthogonal codes. LCD codes have a trivial hull so they are counterparts of self-orthogonal codes. So it is a natural question whether one can embed linear codes into optimal LCD codes. To answer it, we first determine the number of columns to be added to a generator matrix of a linear code in order to embed the given code into an LCD code. Then we characterize all possible forms of shortest LCD embeddings of a linear code. As examples, we start from binary and","authors_text":"Haeun Lim, Ji-Hoon Hong, Jon-Lark Kim, Junmin An","cross_cats":["math.IT"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.IT","submitted_at":"2026-01-28T13:38:17Z","title":"Shortest LCD embeddings of binary, ternary and quaternary linear codes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2601.20600","kind":"arxiv","version":2},"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:b551fb5a77169cc17b05358c3f108670dad0dc63908838ff68c260f35f33bcc7","target":"record","created_at":"2026-06-09T02:07:18Z","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":"9c941f3d703f9dc5195411769994531c10f4a7f9bc36370874ab0b13180f74a1","cross_cats_sorted":["math.IT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.IT","submitted_at":"2026-01-28T13:38:17Z","title_canon_sha256":"537fff4aac1b51b65a0f094614a8a4f30b18ff02ca89b7bc6b84877b3286732b"},"schema_version":"1.0","source":{"id":"2601.20600","kind":"arxiv","version":2}},"canonical_sha256":"96d223c498fcd005b53307195865b885850a2d87c6e90dfeb8361ea3ba988f03","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"96d223c498fcd005b53307195865b885850a2d87c6e90dfeb8361ea3ba988f03","first_computed_at":"2026-06-09T02:07:18.300707Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-09T02:07:18.300707Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qenTAQSStgNEQ6yrFkK77YX5PVWHOWzejeR5GFIc8iQSEHf57ZBJMuY12cEtHl6QSrfxiIB7aMzi18hB3s8UCg==","signature_status":"signed_v1","signed_at":"2026-06-09T02:07:18.301697Z","signed_message":"canonical_sha256_bytes"},"source_id":"2601.20600","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b551fb5a77169cc17b05358c3f108670dad0dc63908838ff68c260f35f33bcc7","sha256:b289769c431c274039e8d32291ccc0df3ab4f0f169591d3ecb6b468e22c0265a"],"state_sha256":"ef4e16552edc745f12b3d43c5cd49c2d578552bba82ac4daef9cab5480adf787"}