{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:YSZ3Q22HMDNYYUV4NGQGQLRKOW","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":"d9bbef13c003319fc212eacaf8e043cd3d106ab7d6ef8b7853714b5391ed289f","cross_cats_sorted":["cs.CC","cs.CG","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-09-22T14:50:25Z","title_canon_sha256":"f1fee4fa9afadce5801d9383288512b211084a5368482cb90b254b06de152e04"},"schema_version":"1.0","source":{"id":"1009.4375","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1009.4375","created_at":"2026-05-18T04:27:03Z"},{"alias_kind":"arxiv_version","alias_value":"1009.4375v2","created_at":"2026-05-18T04:27:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.4375","created_at":"2026-05-18T04:27:03Z"},{"alias_kind":"pith_short_12","alias_value":"YSZ3Q22HMDNY","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_16","alias_value":"YSZ3Q22HMDNYYUV4","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_8","alias_value":"YSZ3Q22H","created_at":"2026-05-18T12:26:17Z"}],"graph_snapshots":[{"event_id":"sha256:e52500aa815a658639fd1aa8db4c84bf0ada9dd7a037b6d74445e03969ec798c","target":"graph","created_at":"2026-05-18T04:27:03Z","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":"A (q,k,t)-design matrix is an m x n matrix whose pattern of zeros/non-zeros satisfies the following design-like condition: each row has at most q non-zeros, each column has at least k non-zeros and the supports of every two columns intersect in at most t rows. We prove that the rank of any (q,k,t)-design matrix over a field of characteristic zero (or sufficiently large finite characteristic) is at least n - (qtn/2k)^2 . Using this result we derive the following applications:\n  (1) Impossibility results for 2-query LCCs over the complex numbers: A 2-query locally correctable code (LCC) is an er","authors_text":"Amir Yehudayoff, Avi Wigderson, Boaz Barak, Zeev Dvir","cross_cats":["cs.CC","cs.CG","math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-09-22T14:50:25Z","title":"Rank Bounds for Design Matrices with Applications to Combinatorial Geometry and Locally Correctable Codes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.4375","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:dbaed8a8dae71bc4bc8c731b8723496da3a39dff465c8a5a2d65c88c740e8787","target":"record","created_at":"2026-05-18T04:27:03Z","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":"d9bbef13c003319fc212eacaf8e043cd3d106ab7d6ef8b7853714b5391ed289f","cross_cats_sorted":["cs.CC","cs.CG","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-09-22T14:50:25Z","title_canon_sha256":"f1fee4fa9afadce5801d9383288512b211084a5368482cb90b254b06de152e04"},"schema_version":"1.0","source":{"id":"1009.4375","kind":"arxiv","version":2}},"canonical_sha256":"c4b3b86b4760db8c52bc69a0682e2a75947dd0fab7f26bcf55edda8b0f8b6e3c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c4b3b86b4760db8c52bc69a0682e2a75947dd0fab7f26bcf55edda8b0f8b6e3c","first_computed_at":"2026-05-18T04:27:03.641507Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:27:03.641507Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"oUWmvhq9z2qPCcZbqxGTJ4cEn+91d61wBO2nsr1VmssYPwlO4maHvU9VPABQkJvgbu8b1KTfGFG3cUh4fe0hDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:27:03.642194Z","signed_message":"canonical_sha256_bytes"},"source_id":"1009.4375","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dbaed8a8dae71bc4bc8c731b8723496da3a39dff465c8a5a2d65c88c740e8787","sha256:e52500aa815a658639fd1aa8db4c84bf0ada9dd7a037b6d74445e03969ec798c"],"state_sha256":"bb5dbaf77c073cc825c6c960bcae4df038232b7454a57c7d6ca3b031d4e91402"}