{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:EZOAJOLI7SW5SWRMRXI2ZFRNYD","short_pith_number":"pith:EZOAJOLI","canonical_record":{"source":{"id":"1507.07855","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-28T17:19:46Z","cross_cats_sorted":["cs.IT","math.IT"],"title_canon_sha256":"5ec821034335d147ad67f19bf991a4e8bbfaaec72e46791c7c065782f83b3f54","abstract_canon_sha256":"7cbd096c3c61cf63a9c1565644a85a814b1b743b63eb3c4be6d14a0b9dd5d429"},"schema_version":"1.0"},"canonical_sha256":"265c04b968fcadd95a2c8dd1ac962dc0ed6cbec908b53a7a327760cab60ced79","source":{"kind":"arxiv","id":"1507.07855","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.07855","created_at":"2026-05-18T00:14:57Z"},{"alias_kind":"arxiv_version","alias_value":"1507.07855v3","created_at":"2026-05-18T00:14:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.07855","created_at":"2026-05-18T00:14:57Z"},{"alias_kind":"pith_short_12","alias_value":"EZOAJOLI7SW5","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_16","alias_value":"EZOAJOLI7SW5SWRM","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_8","alias_value":"EZOAJOLI","created_at":"2026-05-18T12:29:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:EZOAJOLI7SW5SWRMRXI2ZFRNYD","target":"record","payload":{"canonical_record":{"source":{"id":"1507.07855","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-28T17:19:46Z","cross_cats_sorted":["cs.IT","math.IT"],"title_canon_sha256":"5ec821034335d147ad67f19bf991a4e8bbfaaec72e46791c7c065782f83b3f54","abstract_canon_sha256":"7cbd096c3c61cf63a9c1565644a85a814b1b743b63eb3c4be6d14a0b9dd5d429"},"schema_version":"1.0"},"canonical_sha256":"265c04b968fcadd95a2c8dd1ac962dc0ed6cbec908b53a7a327760cab60ced79","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:14:57.566012Z","signature_b64":"RjEE2qUm6/s8aPpKSPAnM7hMcotHMyUF4kbBssHWN0zZcNFH9sPyXzgXvdTzpqSlqjvbWz6A4XDYk4Ozc2cbDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"265c04b968fcadd95a2c8dd1ac962dc0ed6cbec908b53a7a327760cab60ced79","last_reissued_at":"2026-05-18T00:14:57.565290Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:14:57.565290Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1507.07855","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:14:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qLLo8BGqb3L1/v7sqLWglTFustOsuiBu4IYnKXJx+yIPW4/HmzH64HVehZiXzddjIXuTD2N38rc7fZnQttBJBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T01:42:37.917694Z"},"content_sha256":"8a68254e58c00eaf33df4b7411bf8e52f19e1b6582b6d159c5949c7f746c494d","schema_version":"1.0","event_id":"sha256:8a68254e58c00eaf33df4b7411bf8e52f19e1b6582b6d159c5949c7f746c494d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:EZOAJOLI7SW5SWRMRXI2ZFRNYD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Generalized Twisted Gabidulin Codes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.IT"],"primary_cat":"math.CO","authors_text":"Guglielmo Lunardon, Rocco Trombetti, Yue Zhou","submitted_at":"2015-07-28T17:19:46Z","abstract_excerpt":"Let $\\mathcal{C}$ be a set of $m$ by $n$ matrices over $\\mathbb{F}_q$ such that the rank of $A-B$ is at least $d$ for all distinct $A,B\\in \\mathcal{C}$. Suppose that $m\\leqslant n$. If $\\#\\mathcal{C}= q^{n(m-d+1)}$, then $\\mathcal{C}$ is a maximum rank distance (MRD for short) code. Until 2016, there were only two known constructions of MRD codes for arbitrary $1<d<m-1$. One was found by Delsarte (1978) and Gabidulin (1985) independently, and it was later generalized by Kshevetskiy and Gabidulin (2005). We often call them (generalized) Gabidulin codes. Another family was recently obtained by S"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.07855","kind":"arxiv","version":3},"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":"2026-05-18T00:14:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QES7AWG3XA2ZcK3wGJS66mhEizS7hZpVG+LQDpcog40Ll0XcSpq66rxLd002dJp993xCxWA1XagGo3NvDto/Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T01:42:37.918047Z"},"content_sha256":"f5a2b57dfa1551b6e924317d5e103c6f6a1bd2ea4148304b99f0ea9b9b8af11e","schema_version":"1.0","event_id":"sha256:f5a2b57dfa1551b6e924317d5e103c6f6a1bd2ea4148304b99f0ea9b9b8af11e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EZOAJOLI7SW5SWRMRXI2ZFRNYD/bundle.json","state_url":"https://pith.science/pith/EZOAJOLI7SW5SWRMRXI2ZFRNYD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EZOAJOLI7SW5SWRMRXI2ZFRNYD/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":"2026-06-12T01:42:37Z","links":{"resolver":"https://pith.science/pith/EZOAJOLI7SW5SWRMRXI2ZFRNYD","bundle":"https://pith.science/pith/EZOAJOLI7SW5SWRMRXI2ZFRNYD/bundle.json","state":"https://pith.science/pith/EZOAJOLI7SW5SWRMRXI2ZFRNYD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EZOAJOLI7SW5SWRMRXI2ZFRNYD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:EZOAJOLI7SW5SWRMRXI2ZFRNYD","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":"7cbd096c3c61cf63a9c1565644a85a814b1b743b63eb3c4be6d14a0b9dd5d429","cross_cats_sorted":["cs.IT","math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-28T17:19:46Z","title_canon_sha256":"5ec821034335d147ad67f19bf991a4e8bbfaaec72e46791c7c065782f83b3f54"},"schema_version":"1.0","source":{"id":"1507.07855","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.07855","created_at":"2026-05-18T00:14:57Z"},{"alias_kind":"arxiv_version","alias_value":"1507.07855v3","created_at":"2026-05-18T00:14:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.07855","created_at":"2026-05-18T00:14:57Z"},{"alias_kind":"pith_short_12","alias_value":"EZOAJOLI7SW5","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_16","alias_value":"EZOAJOLI7SW5SWRM","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_8","alias_value":"EZOAJOLI","created_at":"2026-05-18T12:29:19Z"}],"graph_snapshots":[{"event_id":"sha256:f5a2b57dfa1551b6e924317d5e103c6f6a1bd2ea4148304b99f0ea9b9b8af11e","target":"graph","created_at":"2026-05-18T00:14:57Z","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":"Let $\\mathcal{C}$ be a set of $m$ by $n$ matrices over $\\mathbb{F}_q$ such that the rank of $A-B$ is at least $d$ for all distinct $A,B\\in \\mathcal{C}$. Suppose that $m\\leqslant n$. If $\\#\\mathcal{C}= q^{n(m-d+1)}$, then $\\mathcal{C}$ is a maximum rank distance (MRD for short) code. Until 2016, there were only two known constructions of MRD codes for arbitrary $1<d<m-1$. One was found by Delsarte (1978) and Gabidulin (1985) independently, and it was later generalized by Kshevetskiy and Gabidulin (2005). We often call them (generalized) Gabidulin codes. Another family was recently obtained by S","authors_text":"Guglielmo Lunardon, Rocco Trombetti, Yue Zhou","cross_cats":["cs.IT","math.IT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-28T17:19:46Z","title":"Generalized Twisted Gabidulin Codes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.07855","kind":"arxiv","version":3},"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":"sha256:8a68254e58c00eaf33df4b7411bf8e52f19e1b6582b6d159c5949c7f746c494d","target":"record","created_at":"2026-05-18T00:14:57Z","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":"7cbd096c3c61cf63a9c1565644a85a814b1b743b63eb3c4be6d14a0b9dd5d429","cross_cats_sorted":["cs.IT","math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-28T17:19:46Z","title_canon_sha256":"5ec821034335d147ad67f19bf991a4e8bbfaaec72e46791c7c065782f83b3f54"},"schema_version":"1.0","source":{"id":"1507.07855","kind":"arxiv","version":3}},"canonical_sha256":"265c04b968fcadd95a2c8dd1ac962dc0ed6cbec908b53a7a327760cab60ced79","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"265c04b968fcadd95a2c8dd1ac962dc0ed6cbec908b53a7a327760cab60ced79","first_computed_at":"2026-05-18T00:14:57.565290Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:14:57.565290Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RjEE2qUm6/s8aPpKSPAnM7hMcotHMyUF4kbBssHWN0zZcNFH9sPyXzgXvdTzpqSlqjvbWz6A4XDYk4Ozc2cbDA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:14:57.566012Z","signed_message":"canonical_sha256_bytes"},"source_id":"1507.07855","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8a68254e58c00eaf33df4b7411bf8e52f19e1b6582b6d159c5949c7f746c494d","sha256:f5a2b57dfa1551b6e924317d5e103c6f6a1bd2ea4148304b99f0ea9b9b8af11e"],"state_sha256":"502fb56ebbe48853666b4240fb06cea3fe18cc93de2da5d533abe1dd723f899f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KabjG1mR9e02xvTJOeQ1LidwZMEBB6wG3gmWwMBQQiBD1FCsBTyHO2YgpHXPh2mNTujT2ZEFSyRtSew3SqmeAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-12T01:42:37.920092Z","bundle_sha256":"a4cbfad2947c6dddf9a2ead053f909267a58f86fbbd35d0b9c2adeabdf014d71"}}