{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:OEU6RYDUBYIODGLPTTKM4OZHCF","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":"5cc9048147cf17b3f9606073c9ae82efd9d6efff4d4acdfa274fe7f3b1cf9e58","cross_cats_sorted":["cs.DM","math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2014-10-21T10:12:42Z","title_canon_sha256":"6c9f681d9d8d7cb76a944f416483160d672b71ef26be59cb27ab1b342ae09fd8"},"schema_version":"1.0","source":{"id":"1410.5604","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.5604","created_at":"2026-05-18T02:39:39Z"},{"alias_kind":"arxiv_version","alias_value":"1410.5604v1","created_at":"2026-05-18T02:39:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.5604","created_at":"2026-05-18T02:39:39Z"},{"alias_kind":"pith_short_12","alias_value":"OEU6RYDUBYIO","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_16","alias_value":"OEU6RYDUBYIODGLP","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_8","alias_value":"OEU6RYDU","created_at":"2026-05-18T12:28:41Z"}],"graph_snapshots":[{"event_id":"sha256:0f7c3e12d27ec7ce92cca5583df5c0f99cda7f371b4fa80b06278c412726b755","target":"graph","created_at":"2026-05-18T02:39:39Z","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 $C$ is a $\\mathbb{Z}_2$-double cyclic code if the set of coordinates can be partitioned into two subsets such that any cyclic shift of the coordinates of both subsets leaves invariant the code. These codes can be identified as submodules of the $\\mathbb{Z}_2[x]$-module $\\mathbb{Z}_2[x]/(x^r-1)\\times\\mathbb{Z}_2[x]/(x^s-1).$ We determine the structure of $\\mathbb{Z}_2$-double cyclic codes giving the generator polynomials of these codes. The related polynomial representation of $\\mathbb{Z}_2$-double cyclic codes and its duals, and the relations between the polynomial generat","authors_text":"Cristina Fern\\'andez-C\\'ordoba, Joaquim Borges, Roger Ten-Valls","cross_cats":["cs.DM","math.IT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2014-10-21T10:12:42Z","title":"Z2-double cyclic codes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.5604","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":"sha256:1317fbdcd4114d24e811138112b0d6911e6239669bcd11b683be18144bfeb0f1","target":"record","created_at":"2026-05-18T02:39:39Z","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":"5cc9048147cf17b3f9606073c9ae82efd9d6efff4d4acdfa274fe7f3b1cf9e58","cross_cats_sorted":["cs.DM","math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2014-10-21T10:12:42Z","title_canon_sha256":"6c9f681d9d8d7cb76a944f416483160d672b71ef26be59cb27ab1b342ae09fd8"},"schema_version":"1.0","source":{"id":"1410.5604","kind":"arxiv","version":1}},"canonical_sha256":"7129e8e0740e10e1996f9cd4ce3b271172c0aaf9b8bd41ce10e99600980bb30b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7129e8e0740e10e1996f9cd4ce3b271172c0aaf9b8bd41ce10e99600980bb30b","first_computed_at":"2026-05-18T02:39:39.472053Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:39:39.472053Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"nUITCEsZe923Bnw/RpPh2qlOdWckOgZFbtVAyN3+qBBJWV9ytGZum+eqjp7Ezk7l+KhbCTTljAPH/3CjbFu1Aw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:39:39.472516Z","signed_message":"canonical_sha256_bytes"},"source_id":"1410.5604","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1317fbdcd4114d24e811138112b0d6911e6239669bcd11b683be18144bfeb0f1","sha256:0f7c3e12d27ec7ce92cca5583df5c0f99cda7f371b4fa80b06278c412726b755"],"state_sha256":"917879d2d1a4d76c0243816a8f473ef5a3e9595bb9fff79d60ebe4d3b193662a"}