{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:Q5KT6CAF3LE4C3NGB57QB3KTFO","short_pith_number":"pith:Q5KT6CAF","canonical_record":{"source":{"id":"1307.3047","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-07-11T10:39:45Z","cross_cats_sorted":["cs.IT","math.IT"],"title_canon_sha256":"badce1cc52c8f379407310c0cae777ac9fd0d415be81703a90ad19d9cc432297","abstract_canon_sha256":"74aae24b507137bfd25291cf3037e883d3de596b9ef2c612946436d336182eab"},"schema_version":"1.0"},"canonical_sha256":"87553f0805dac9c16da60f7f00ed532ba7d383ad4e4512c7724d23cbda58d632","source":{"kind":"arxiv","id":"1307.3047","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.3047","created_at":"2026-05-18T03:18:41Z"},{"alias_kind":"arxiv_version","alias_value":"1307.3047v1","created_at":"2026-05-18T03:18:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.3047","created_at":"2026-05-18T03:18:41Z"},{"alias_kind":"pith_short_12","alias_value":"Q5KT6CAF3LE4","created_at":"2026-05-18T12:27:57Z"},{"alias_kind":"pith_short_16","alias_value":"Q5KT6CAF3LE4C3NG","created_at":"2026-05-18T12:27:57Z"},{"alias_kind":"pith_short_8","alias_value":"Q5KT6CAF","created_at":"2026-05-18T12:27:57Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:Q5KT6CAF3LE4C3NGB57QB3KTFO","target":"record","payload":{"canonical_record":{"source":{"id":"1307.3047","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-07-11T10:39:45Z","cross_cats_sorted":["cs.IT","math.IT"],"title_canon_sha256":"badce1cc52c8f379407310c0cae777ac9fd0d415be81703a90ad19d9cc432297","abstract_canon_sha256":"74aae24b507137bfd25291cf3037e883d3de596b9ef2c612946436d336182eab"},"schema_version":"1.0"},"canonical_sha256":"87553f0805dac9c16da60f7f00ed532ba7d383ad4e4512c7724d23cbda58d632","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:18:41.421774Z","signature_b64":"5r8hiuODIbNxOL44ePSvunLzU6Oa6ftYrEtqiQ1mDyGSSPpUUM+qFzb4VIKo9adOePOnYN+PuQSBF8HnWn1aCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"87553f0805dac9c16da60f7f00ed532ba7d383ad4e4512c7724d23cbda58d632","last_reissued_at":"2026-05-18T03:18:41.421036Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:18:41.421036Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1307.3047","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:18:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RCHs9ZRbS7vQpOaO0kt7np1NdPl1gLqF5Cca8gE5SFTX7pAQPp1VCStewkVqhH3B2Ems+3KZfCsl8iqR4W+JDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T06:44:00.211308Z"},"content_sha256":"e0bd17187c8a8802eeaf3ec5dfb6dbaccf24ea2621db82970e51d72f60ff0bf4","schema_version":"1.0","event_id":"sha256:e0bd17187c8a8802eeaf3ec5dfb6dbaccf24ea2621db82970e51d72f60ff0bf4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:Q5KT6CAF3LE4C3NGB57QB3KTFO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Linear Codes over Z_4+uZ_4: MacWilliams identities, projections, and formally self-dual codes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.IT"],"primary_cat":"math.RA","authors_text":"Bahattin Yildiz, Suat Karadeniz","submitted_at":"2013-07-11T10:39:45Z","abstract_excerpt":"Linear codes are considered over the ring Z_4+uZ_4, a non-chain extension of Z_4. Lee weights, Gray maps for these codes are defined and MacWilliams identities for the complete, symmetrized and Lee weight enumerators are proved. Two projections from Z_4+uZ_4 to the rings Z_4 and F_2+uF_2 are considered and self-dual codes over Z_4+uZ_4 are studied in connection with these projections. Finally three constructions are given for formally self-dual codes over Z_4+uZ_4 and their Z_4-images together with some good examples of formally self-dual Z_4-codes obtained through these constructions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.3047","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:18:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5VMLhwmmsYP1PFnc5GF5JoF6+rAhxIBswsZVEveLUdpYFln+7BdYlKwJr6n84aHBYxm7lbv4W/TkDSvpEccMBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T06:44:00.212064Z"},"content_sha256":"3fd18423bab95acfcdc48634cd65c373a8a13719e0b89be2462c998c3c37dbdc","schema_version":"1.0","event_id":"sha256:3fd18423bab95acfcdc48634cd65c373a8a13719e0b89be2462c998c3c37dbdc"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Q5KT6CAF3LE4C3NGB57QB3KTFO/bundle.json","state_url":"https://pith.science/pith/Q5KT6CAF3LE4C3NGB57QB3KTFO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Q5KT6CAF3LE4C3NGB57QB3KTFO/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-09T06:44:00Z","links":{"resolver":"https://pith.science/pith/Q5KT6CAF3LE4C3NGB57QB3KTFO","bundle":"https://pith.science/pith/Q5KT6CAF3LE4C3NGB57QB3KTFO/bundle.json","state":"https://pith.science/pith/Q5KT6CAF3LE4C3NGB57QB3KTFO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Q5KT6CAF3LE4C3NGB57QB3KTFO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:Q5KT6CAF3LE4C3NGB57QB3KTFO","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":"74aae24b507137bfd25291cf3037e883d3de596b9ef2c612946436d336182eab","cross_cats_sorted":["cs.IT","math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-07-11T10:39:45Z","title_canon_sha256":"badce1cc52c8f379407310c0cae777ac9fd0d415be81703a90ad19d9cc432297"},"schema_version":"1.0","source":{"id":"1307.3047","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.3047","created_at":"2026-05-18T03:18:41Z"},{"alias_kind":"arxiv_version","alias_value":"1307.3047v1","created_at":"2026-05-18T03:18:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.3047","created_at":"2026-05-18T03:18:41Z"},{"alias_kind":"pith_short_12","alias_value":"Q5KT6CAF3LE4","created_at":"2026-05-18T12:27:57Z"},{"alias_kind":"pith_short_16","alias_value":"Q5KT6CAF3LE4C3NG","created_at":"2026-05-18T12:27:57Z"},{"alias_kind":"pith_short_8","alias_value":"Q5KT6CAF","created_at":"2026-05-18T12:27:57Z"}],"graph_snapshots":[{"event_id":"sha256:3fd18423bab95acfcdc48634cd65c373a8a13719e0b89be2462c998c3c37dbdc","target":"graph","created_at":"2026-05-18T03:18:41Z","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":"Linear codes are considered over the ring Z_4+uZ_4, a non-chain extension of Z_4. Lee weights, Gray maps for these codes are defined and MacWilliams identities for the complete, symmetrized and Lee weight enumerators are proved. Two projections from Z_4+uZ_4 to the rings Z_4 and F_2+uF_2 are considered and self-dual codes over Z_4+uZ_4 are studied in connection with these projections. Finally three constructions are given for formally self-dual codes over Z_4+uZ_4 and their Z_4-images together with some good examples of formally self-dual Z_4-codes obtained through these constructions.","authors_text":"Bahattin Yildiz, Suat Karadeniz","cross_cats":["cs.IT","math.IT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-07-11T10:39:45Z","title":"Linear Codes over Z_4+uZ_4: MacWilliams identities, projections, and formally self-dual codes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.3047","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:e0bd17187c8a8802eeaf3ec5dfb6dbaccf24ea2621db82970e51d72f60ff0bf4","target":"record","created_at":"2026-05-18T03:18:41Z","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":"74aae24b507137bfd25291cf3037e883d3de596b9ef2c612946436d336182eab","cross_cats_sorted":["cs.IT","math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-07-11T10:39:45Z","title_canon_sha256":"badce1cc52c8f379407310c0cae777ac9fd0d415be81703a90ad19d9cc432297"},"schema_version":"1.0","source":{"id":"1307.3047","kind":"arxiv","version":1}},"canonical_sha256":"87553f0805dac9c16da60f7f00ed532ba7d383ad4e4512c7724d23cbda58d632","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"87553f0805dac9c16da60f7f00ed532ba7d383ad4e4512c7724d23cbda58d632","first_computed_at":"2026-05-18T03:18:41.421036Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:18:41.421036Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5r8hiuODIbNxOL44ePSvunLzU6Oa6ftYrEtqiQ1mDyGSSPpUUM+qFzb4VIKo9adOePOnYN+PuQSBF8HnWn1aCw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:18:41.421774Z","signed_message":"canonical_sha256_bytes"},"source_id":"1307.3047","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e0bd17187c8a8802eeaf3ec5dfb6dbaccf24ea2621db82970e51d72f60ff0bf4","sha256:3fd18423bab95acfcdc48634cd65c373a8a13719e0b89be2462c998c3c37dbdc"],"state_sha256":"6732073d7d1636d18a046b03efea580a8f7188637a8f227a262989deca830816"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wJwcvYm96c9wOkxCpg7oGCBufELk6c0R5OVpUHM+GYQrOMSYBMRukHSaG2/i8BmClpug5/qPjQ9clUAPT8XaCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T06:44:00.216449Z","bundle_sha256":"c51ada14283659b527bd5ad02036a14427c7190a67591d5c2ac0ead17bf6f376"}}