{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:OKU2AR7NASWRSQNQGX5N7OZPYQ","short_pith_number":"pith:OKU2AR7N","schema_version":"1.0","canonical_sha256":"72a9a047ed04ad1941b035fadfbb2fc428fbdf945ef0ae13d8b891be5f4c394a","source":{"kind":"arxiv","id":"1211.5251","version":1},"attestation_state":"computed","paper":{"title":"Families of Hadamard Z2Z4Q8-codes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.IT"],"primary_cat":"cs.IT","authors_text":"\\'Angel del Rio, Josep Rif\\`a","submitted_at":"2012-11-22T10:56:58Z","abstract_excerpt":"A Z2Z4Q8-code is a non-empty subgroup of a direct product of copies of Z_2, Z_4 and Q_8 (the binary field, the ring of integers modulo 4 and the quaternion group on eight elements, respectively). Such Z2Z4Q8-codes are translation invariant propelinear codes as the well known Z_4-linear or Z_2Z_4-linear codes.\n  In the current paper, we show that there exist \"pure\" Z2Z4Q8-codes, that is, codes that do not admit any abelian translation invariant propelinear structure. We study the dimension of the kernel and rank of the Z2Z4Q8-codes, and we give upper and lower bounds for these parameters. We gi"},"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":"1211.5251","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2012-11-22T10:56:58Z","cross_cats_sorted":["math.CO","math.IT"],"title_canon_sha256":"c773ce9d5e1238a217b2ef76672b4bb9188fe8986b52d631a76328a14f78a598","abstract_canon_sha256":"19973e83341b902375f767ca6a027852f4cebc3620d1c6be775da63e13ebdedf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:40:04.894813Z","signature_b64":"E4IuIygZROVtPHwCL6XRzwYkf77BaKQm8zRjG6lQS6lz55jac/Jkc2ERWid5dk5eWYrpoEk98+8UMs2O7pyICg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"72a9a047ed04ad1941b035fadfbb2fc428fbdf945ef0ae13d8b891be5f4c394a","last_reissued_at":"2026-05-18T03:40:04.894281Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:40:04.894281Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Families of Hadamard Z2Z4Q8-codes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.IT"],"primary_cat":"cs.IT","authors_text":"\\'Angel del Rio, Josep Rif\\`a","submitted_at":"2012-11-22T10:56:58Z","abstract_excerpt":"A Z2Z4Q8-code is a non-empty subgroup of a direct product of copies of Z_2, Z_4 and Q_8 (the binary field, the ring of integers modulo 4 and the quaternion group on eight elements, respectively). Such Z2Z4Q8-codes are translation invariant propelinear codes as the well known Z_4-linear or Z_2Z_4-linear codes.\n  In the current paper, we show that there exist \"pure\" Z2Z4Q8-codes, that is, codes that do not admit any abelian translation invariant propelinear structure. We study the dimension of the kernel and rank of the Z2Z4Q8-codes, and we give upper and lower bounds for these parameters. We gi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.5251","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":"1211.5251","created_at":"2026-05-18T03:40:04.894373+00:00"},{"alias_kind":"arxiv_version","alias_value":"1211.5251v1","created_at":"2026-05-18T03:40:04.894373+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.5251","created_at":"2026-05-18T03:40:04.894373+00:00"},{"alias_kind":"pith_short_12","alias_value":"OKU2AR7NASWR","created_at":"2026-05-18T12:27:16.716162+00:00"},{"alias_kind":"pith_short_16","alias_value":"OKU2AR7NASWRSQNQ","created_at":"2026-05-18T12:27:16.716162+00:00"},{"alias_kind":"pith_short_8","alias_value":"OKU2AR7N","created_at":"2026-05-18T12:27:16.716162+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/OKU2AR7NASWRSQNQGX5N7OZPYQ","json":"https://pith.science/pith/OKU2AR7NASWRSQNQGX5N7OZPYQ.json","graph_json":"https://pith.science/api/pith-number/OKU2AR7NASWRSQNQGX5N7OZPYQ/graph.json","events_json":"https://pith.science/api/pith-number/OKU2AR7NASWRSQNQGX5N7OZPYQ/events.json","paper":"https://pith.science/paper/OKU2AR7N"},"agent_actions":{"view_html":"https://pith.science/pith/OKU2AR7NASWRSQNQGX5N7OZPYQ","download_json":"https://pith.science/pith/OKU2AR7NASWRSQNQGX5N7OZPYQ.json","view_paper":"https://pith.science/paper/OKU2AR7N","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1211.5251&json=true","fetch_graph":"https://pith.science/api/pith-number/OKU2AR7NASWRSQNQGX5N7OZPYQ/graph.json","fetch_events":"https://pith.science/api/pith-number/OKU2AR7NASWRSQNQGX5N7OZPYQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OKU2AR7NASWRSQNQGX5N7OZPYQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OKU2AR7NASWRSQNQGX5N7OZPYQ/action/storage_attestation","attest_author":"https://pith.science/pith/OKU2AR7NASWRSQNQGX5N7OZPYQ/action/author_attestation","sign_citation":"https://pith.science/pith/OKU2AR7NASWRSQNQGX5N7OZPYQ/action/citation_signature","submit_replication":"https://pith.science/pith/OKU2AR7NASWRSQNQGX5N7OZPYQ/action/replication_record"}},"created_at":"2026-05-18T03:40:04.894373+00:00","updated_at":"2026-05-18T03:40:04.894373+00:00"}