{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:DFX4XIDWXJ3HMD2JASJAUHHAWD","short_pith_number":"pith:DFX4XIDW","canonical_record":{"source":{"id":"1803.02523","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2018-03-07T04:59:32Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"4bc438cda0c1631125970a87afc3aebf106a2faa35fd27c10f7077cbedca74ef","abstract_canon_sha256":"79db3d1545542e6c2bf3fceef033548ec3eadb3dddab5d05b50d12f17ea1b5c8"},"schema_version":"1.0"},"canonical_sha256":"196fcba076ba76760f4904920a1ce0b0e3fe313ff5143c1fb725d2dec015322b","source":{"kind":"arxiv","id":"1803.02523","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1803.02523","created_at":"2026-05-18T00:20:29Z"},{"alias_kind":"arxiv_version","alias_value":"1803.02523v2","created_at":"2026-05-18T00:20:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.02523","created_at":"2026-05-18T00:20:29Z"},{"alias_kind":"pith_short_12","alias_value":"DFX4XIDWXJ3H","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_16","alias_value":"DFX4XIDWXJ3HMD2J","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_8","alias_value":"DFX4XIDW","created_at":"2026-05-18T12:32:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:DFX4XIDWXJ3HMD2JASJAUHHAWD","target":"record","payload":{"canonical_record":{"source":{"id":"1803.02523","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2018-03-07T04:59:32Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"4bc438cda0c1631125970a87afc3aebf106a2faa35fd27c10f7077cbedca74ef","abstract_canon_sha256":"79db3d1545542e6c2bf3fceef033548ec3eadb3dddab5d05b50d12f17ea1b5c8"},"schema_version":"1.0"},"canonical_sha256":"196fcba076ba76760f4904920a1ce0b0e3fe313ff5143c1fb725d2dec015322b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:20:29.780168Z","signature_b64":"BSXXKsgA01A7MeB2ZgGZPZX2hz9J3XS5hdVTC6xIMF48Ck5eYH5xMPxyk2XxQQnXxwyrMjs31HjU/O1JO6QiCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"196fcba076ba76760f4904920a1ce0b0e3fe313ff5143c1fb725d2dec015322b","last_reissued_at":"2026-05-18T00:20:29.779608Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:20:29.779608Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1803.02523","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:20:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GtTsgSPuYprIHgTyHgh7+V8LjTJlZGtStC49LyGkuc8g6E/aEE76W019gX1SOi30SBTVAonCGFUdUG6+xf9NCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T21:08:04.954175Z"},"content_sha256":"74e053185d36257619e228a373332c56a278efd172046b20726a7c7e974fc6af","schema_version":"1.0","event_id":"sha256:74e053185d36257619e228a373332c56a278efd172046b20726a7c7e974fc6af"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:DFX4XIDWXJ3HMD2JASJAUHHAWD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"MDS matrices over small fields: A proof of the GM-MDS conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Shachar Lovett","submitted_at":"2018-03-07T04:59:32Z","abstract_excerpt":"An MDS matrix is a matrix whose minors all have full rank. A question arising in coding theory is what zero patterns can MDS matrices have. There is a natural combinatorial characterization (called the MDS condition) which is necessary over any field, as well as sufficient over very large fields by a probabilistic argument.\n  Dau et al. (ISIT 2014) conjectured that the MDS condition is sufficient over small fields as well, where the construction of the matrix is algebraic instead of probabilistic. This is known as the GM-MDS conjecture. Concretely, if a $k \\times n$ zero pattern satisfies the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.02523","kind":"arxiv","version":2},"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":"2026-05-18T00:20:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wvcW3Mlj+4CgC0SZ1ZnqOeDKPAHayMKs9I9sRKEP542cwp+InCUNXe8vyspv9OSZDG+JLKltGkwDmifKxLO/Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T21:08:04.954960Z"},"content_sha256":"0b2b47ad03059dc369467ed5073132f4f1fb1860167ebfb16800ef918a36a310","schema_version":"1.0","event_id":"sha256:0b2b47ad03059dc369467ed5073132f4f1fb1860167ebfb16800ef918a36a310"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DFX4XIDWXJ3HMD2JASJAUHHAWD/bundle.json","state_url":"https://pith.science/pith/DFX4XIDWXJ3HMD2JASJAUHHAWD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DFX4XIDWXJ3HMD2JASJAUHHAWD/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":"2026-06-10T21:08:04Z","links":{"resolver":"https://pith.science/pith/DFX4XIDWXJ3HMD2JASJAUHHAWD","bundle":"https://pith.science/pith/DFX4XIDWXJ3HMD2JASJAUHHAWD/bundle.json","state":"https://pith.science/pith/DFX4XIDWXJ3HMD2JASJAUHHAWD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DFX4XIDWXJ3HMD2JASJAUHHAWD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:DFX4XIDWXJ3HMD2JASJAUHHAWD","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":"79db3d1545542e6c2bf3fceef033548ec3eadb3dddab5d05b50d12f17ea1b5c8","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2018-03-07T04:59:32Z","title_canon_sha256":"4bc438cda0c1631125970a87afc3aebf106a2faa35fd27c10f7077cbedca74ef"},"schema_version":"1.0","source":{"id":"1803.02523","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1803.02523","created_at":"2026-05-18T00:20:29Z"},{"alias_kind":"arxiv_version","alias_value":"1803.02523v2","created_at":"2026-05-18T00:20:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.02523","created_at":"2026-05-18T00:20:29Z"},{"alias_kind":"pith_short_12","alias_value":"DFX4XIDWXJ3H","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_16","alias_value":"DFX4XIDWXJ3HMD2J","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_8","alias_value":"DFX4XIDW","created_at":"2026-05-18T12:32:19Z"}],"graph_snapshots":[{"event_id":"sha256:0b2b47ad03059dc369467ed5073132f4f1fb1860167ebfb16800ef918a36a310","target":"graph","created_at":"2026-05-18T00:20:29Z","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":"An MDS matrix is a matrix whose minors all have full rank. A question arising in coding theory is what zero patterns can MDS matrices have. There is a natural combinatorial characterization (called the MDS condition) which is necessary over any field, as well as sufficient over very large fields by a probabilistic argument.\n  Dau et al. (ISIT 2014) conjectured that the MDS condition is sufficient over small fields as well, where the construction of the matrix is algebraic instead of probabilistic. This is known as the GM-MDS conjecture. Concretely, if a $k \\times n$ zero pattern satisfies the ","authors_text":"Shachar Lovett","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2018-03-07T04:59:32Z","title":"MDS matrices over small fields: A proof of the GM-MDS conjecture"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.02523","kind":"arxiv","version":2},"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":"sha256:74e053185d36257619e228a373332c56a278efd172046b20726a7c7e974fc6af","target":"record","created_at":"2026-05-18T00:20:29Z","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":"79db3d1545542e6c2bf3fceef033548ec3eadb3dddab5d05b50d12f17ea1b5c8","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2018-03-07T04:59:32Z","title_canon_sha256":"4bc438cda0c1631125970a87afc3aebf106a2faa35fd27c10f7077cbedca74ef"},"schema_version":"1.0","source":{"id":"1803.02523","kind":"arxiv","version":2}},"canonical_sha256":"196fcba076ba76760f4904920a1ce0b0e3fe313ff5143c1fb725d2dec015322b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"196fcba076ba76760f4904920a1ce0b0e3fe313ff5143c1fb725d2dec015322b","first_computed_at":"2026-05-18T00:20:29.779608Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:20:29.779608Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BSXXKsgA01A7MeB2ZgGZPZX2hz9J3XS5hdVTC6xIMF48Ck5eYH5xMPxyk2XxQQnXxwyrMjs31HjU/O1JO6QiCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:20:29.780168Z","signed_message":"canonical_sha256_bytes"},"source_id":"1803.02523","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:74e053185d36257619e228a373332c56a278efd172046b20726a7c7e974fc6af","sha256:0b2b47ad03059dc369467ed5073132f4f1fb1860167ebfb16800ef918a36a310"],"state_sha256":"13d13298ec34c3d797142df53cf80fc8541585644907635351e9bcbcb668ddbc"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Txzz4jZRnRvFIE/PPV6wJM+JUNU40E0Zrx2QnqXE+nUAZFpBK2xrBSfbXHi2b5vUrnwI1Dcu051fh1V8MZyPDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T21:08:04.959257Z","bundle_sha256":"cdf6ba47295837b35f32bbf2fc5d22d5fe60fd30e4954ef10f6d9d84d1ba449f"}}