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We also exhibit an example of a $\\Z_2$-linear code which is not $\\Z_2\\Z_2[u]$-linear. Also, we state that duality of $\\Z_2\\Z_2[u]$-linear codes is the same that duality of $\\Z_2$-linear codes.\n  Finally, we prove that the class of $\\Z_2\\Z_4$-linear codes which are also $\\Z_2$-linear is strictly contained in the class of $\\Z_2\\Z_2[u"},"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":"1611.05655","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-11-17T12:27:33Z","cross_cats_sorted":[],"title_canon_sha256":"704b2d4748b10ec2adfcc975c16f735198d4bf7ab6c037254c01238da2a484f0","abstract_canon_sha256":"a1bc93450abf20d1589706d62d38574bdf6c0adc28d3e3f18da8e43ba61337b2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:58:02.970338Z","signature_b64":"AHZjUKyr91DvVafcNVc5CpVXQMuFP3vpZBAOI1TjTo9NmJ4327da5fD1CaOuvz6yg9e4EThSrpnnyxKQR1KRCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3c55c6c0150f38da59133d811f2ef3e2b04843b2b8f4ce30cb7b6d5bc875558c","last_reissued_at":"2026-05-18T00:58:02.969807Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:58:02.969807Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A characterization of $\\mathbb{Z}_2\\mathbb{Z}_2[u]$-linear codes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Joaquim Borges","submitted_at":"2016-11-17T12:27:33Z","abstract_excerpt":"We prove that the class of $\\Z_2\\Z_2[u]$-linear codes is exactly the class of $\\Z_2$-linear codes with automorphism group of even order. Using this characterization, we give examples of known codes, e.g. perfect codes, which has a nontrivial $\\Z_2\\Z_2[u]$ structure. We also exhibit an example of a $\\Z_2$-linear code which is not $\\Z_2\\Z_2[u]$-linear. Also, we state that duality of $\\Z_2\\Z_2[u]$-linear codes is the same that duality of $\\Z_2$-linear codes.\n  Finally, we prove that the class of $\\Z_2\\Z_4$-linear codes which are also $\\Z_2$-linear is strictly contained in the class of $\\Z_2\\Z_2[u"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.05655","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"1611.05655","created_at":"2026-05-18T00:58:02.969891+00:00"},{"alias_kind":"arxiv_version","alias_value":"1611.05655v1","created_at":"2026-05-18T00:58:02.969891+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.05655","created_at":"2026-05-18T00:58:02.969891+00:00"},{"alias_kind":"pith_short_12","alias_value":"HRK4NQAVB44N","created_at":"2026-05-18T12:30:19.053100+00:00"},{"alias_kind":"pith_short_16","alias_value":"HRK4NQAVB44NUWIT","created_at":"2026-05-18T12:30:19.053100+00:00"},{"alias_kind":"pith_short_8","alias_value":"HRK4NQAV","created_at":"2026-05-18T12:30:19.053100+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/HRK4NQAVB44NUWITHWAR6LXT4K","json":"https://pith.science/pith/HRK4NQAVB44NUWITHWAR6LXT4K.json","graph_json":"https://pith.science/api/pith-number/HRK4NQAVB44NUWITHWAR6LXT4K/graph.json","events_json":"https://pith.science/api/pith-number/HRK4NQAVB44NUWITHWAR6LXT4K/events.json","paper":"https://pith.science/paper/HRK4NQAV"},"agent_actions":{"view_html":"https://pith.science/pith/HRK4NQAVB44NUWITHWAR6LXT4K","download_json":"https://pith.science/pith/HRK4NQAVB44NUWITHWAR6LXT4K.json","view_paper":"https://pith.science/paper/HRK4NQAV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1611.05655&json=true","fetch_graph":"https://pith.science/api/pith-number/HRK4NQAVB44NUWITHWAR6LXT4K/graph.json","fetch_events":"https://pith.science/api/pith-number/HRK4NQAVB44NUWITHWAR6LXT4K/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HRK4NQAVB44NUWITHWAR6LXT4K/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HRK4NQAVB44NUWITHWAR6LXT4K/action/storage_attestation","attest_author":"https://pith.science/pith/HRK4NQAVB44NUWITHWAR6LXT4K/action/author_attestation","sign_citation":"https://pith.science/pith/HRK4NQAVB44NUWITHWAR6LXT4K/action/citation_signature","submit_replication":"https://pith.science/pith/HRK4NQAVB44NUWITHWAR6LXT4K/action/replication_record"}},"created_at":"2026-05-18T00:58:02.969891+00:00","updated_at":"2026-05-18T00:58:02.969891+00:00"}