{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:3KCTPG5PQLZXOIXVGT62CSVB3N","short_pith_number":"pith:3KCTPG5P","canonical_record":{"source":{"id":"1304.5010","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2013-04-18T03:17:23Z","cross_cats_sorted":["math.CO","math.GR","math.RT"],"title_canon_sha256":"6838c147b890e4eea1c393491e2a6824c9ad0e780b52fe34d598b53da2b02703","abstract_canon_sha256":"2613b117609f2945cc5b1306a3f6c41059ffac800a5af90b836a7d9b28c7439b"},"schema_version":"1.0"},"canonical_sha256":"da85379baf82f37722f534fda14aa1db76ae330a9ec2a18393ff3bb44b40a0b0","source":{"kind":"arxiv","id":"1304.5010","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1304.5010","created_at":"2026-05-18T03:26:51Z"},{"alias_kind":"arxiv_version","alias_value":"1304.5010v4","created_at":"2026-05-18T03:26:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.5010","created_at":"2026-05-18T03:26:51Z"},{"alias_kind":"pith_short_12","alias_value":"3KCTPG5PQLZX","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_16","alias_value":"3KCTPG5PQLZXOIXV","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_8","alias_value":"3KCTPG5P","created_at":"2026-05-18T12:27:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:3KCTPG5PQLZXOIXVGT62CSVB3N","target":"record","payload":{"canonical_record":{"source":{"id":"1304.5010","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2013-04-18T03:17:23Z","cross_cats_sorted":["math.CO","math.GR","math.RT"],"title_canon_sha256":"6838c147b890e4eea1c393491e2a6824c9ad0e780b52fe34d598b53da2b02703","abstract_canon_sha256":"2613b117609f2945cc5b1306a3f6c41059ffac800a5af90b836a7d9b28c7439b"},"schema_version":"1.0"},"canonical_sha256":"da85379baf82f37722f534fda14aa1db76ae330a9ec2a18393ff3bb44b40a0b0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:26:51.997349Z","signature_b64":"9RSR1LkY7Zv/q6M45r5ZlJt8Rhp4vYbXdJ2nZO/jL88KP42KiK1pTbVqaONKovWOrZ5upMjNiRl9wQMtPdC2AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"da85379baf82f37722f534fda14aa1db76ae330a9ec2a18393ff3bb44b40a0b0","last_reissued_at":"2026-05-18T03:26:51.996689Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:26:51.996689Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1304.5010","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:26:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"E2OkiMuLIG6TVlVUjpNBZlh4EcH9RA5ozWwIA/9hj+kMA/OxWAN0nPdVqy6epUYAD17H5kazuGYJGkChPypIAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-20T15:19:07.881704Z"},"content_sha256":"6d785c943cef6654d382df7ee10078f3373af6c062fc031f29dc3daff4d4ba34","schema_version":"1.0","event_id":"sha256:6d785c943cef6654d382df7ee10078f3373af6c062fc031f29dc3daff4d4ba34"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:3KCTPG5PQLZXOIXVGT62CSVB3N","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Small-Bias Sets for Nonabelian Groups: Derandomizing the Alon-Roichman Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.GR","math.RT"],"primary_cat":"cs.CC","authors_text":"Alexander Russell, Cristopher Moore, Sixia Chen","submitted_at":"2013-04-18T03:17:23Z","abstract_excerpt":"In analogy with epsilon-biased sets over Z_2^n, we construct explicit epsilon-biased sets over nonabelian finite groups G. That is, we find sets S subset G such that | Exp_{x in S} rho(x)| <= epsilon for any nontrivial irreducible representation rho. Equivalently, such sets make G's Cayley graph an expander with eigenvalue |lambda| <= epsilon. The Alon-Roichman theorem shows that random sets of size O(log |G| / epsilon^2) suffice. For groups of the form G = G_1 x ... x G_n, our construction has size poly(max_i |G_i|, n, epsilon^{-1}), and we show that a set S \\subset G^n considered by Meka and"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.5010","kind":"arxiv","version":4},"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":"2026-05-18T03:26:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7sOE/ajqoL94Rx5mRaajl6ehzXi0LBFK5RdYLpyX8JiOflhJmAfmGZxxk4HM6vgZtbYvAoDik1lBW0LbmOAWAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-20T15:19:07.882306Z"},"content_sha256":"77499219faf2ce8883747fc20ef953b2b19648a9ed58c223e7c76b3c1776512b","schema_version":"1.0","event_id":"sha256:77499219faf2ce8883747fc20ef953b2b19648a9ed58c223e7c76b3c1776512b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3KCTPG5PQLZXOIXVGT62CSVB3N/bundle.json","state_url":"https://pith.science/pith/3KCTPG5PQLZXOIXVGT62CSVB3N/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3KCTPG5PQLZXOIXVGT62CSVB3N/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":"2026-05-20T15:19:07Z","links":{"resolver":"https://pith.science/pith/3KCTPG5PQLZXOIXVGT62CSVB3N","bundle":"https://pith.science/pith/3KCTPG5PQLZXOIXVGT62CSVB3N/bundle.json","state":"https://pith.science/pith/3KCTPG5PQLZXOIXVGT62CSVB3N/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3KCTPG5PQLZXOIXVGT62CSVB3N/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:3KCTPG5PQLZXOIXVGT62CSVB3N","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":"2613b117609f2945cc5b1306a3f6c41059ffac800a5af90b836a7d9b28c7439b","cross_cats_sorted":["math.CO","math.GR","math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2013-04-18T03:17:23Z","title_canon_sha256":"6838c147b890e4eea1c393491e2a6824c9ad0e780b52fe34d598b53da2b02703"},"schema_version":"1.0","source":{"id":"1304.5010","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1304.5010","created_at":"2026-05-18T03:26:51Z"},{"alias_kind":"arxiv_version","alias_value":"1304.5010v4","created_at":"2026-05-18T03:26:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.5010","created_at":"2026-05-18T03:26:51Z"},{"alias_kind":"pith_short_12","alias_value":"3KCTPG5PQLZX","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_16","alias_value":"3KCTPG5PQLZXOIXV","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_8","alias_value":"3KCTPG5P","created_at":"2026-05-18T12:27:32Z"}],"graph_snapshots":[{"event_id":"sha256:77499219faf2ce8883747fc20ef953b2b19648a9ed58c223e7c76b3c1776512b","target":"graph","created_at":"2026-05-18T03:26:51Z","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":"In analogy with epsilon-biased sets over Z_2^n, we construct explicit epsilon-biased sets over nonabelian finite groups G. That is, we find sets S subset G such that | Exp_{x in S} rho(x)| <= epsilon for any nontrivial irreducible representation rho. Equivalently, such sets make G's Cayley graph an expander with eigenvalue |lambda| <= epsilon. The Alon-Roichman theorem shows that random sets of size O(log |G| / epsilon^2) suffice. For groups of the form G = G_1 x ... x G_n, our construction has size poly(max_i |G_i|, n, epsilon^{-1}), and we show that a set S \\subset G^n considered by Meka and","authors_text":"Alexander Russell, Cristopher Moore, Sixia Chen","cross_cats":["math.CO","math.GR","math.RT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2013-04-18T03:17:23Z","title":"Small-Bias Sets for Nonabelian Groups: Derandomizing the Alon-Roichman Theorem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.5010","kind":"arxiv","version":4},"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":"sha256:6d785c943cef6654d382df7ee10078f3373af6c062fc031f29dc3daff4d4ba34","target":"record","created_at":"2026-05-18T03:26:51Z","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":"2613b117609f2945cc5b1306a3f6c41059ffac800a5af90b836a7d9b28c7439b","cross_cats_sorted":["math.CO","math.GR","math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2013-04-18T03:17:23Z","title_canon_sha256":"6838c147b890e4eea1c393491e2a6824c9ad0e780b52fe34d598b53da2b02703"},"schema_version":"1.0","source":{"id":"1304.5010","kind":"arxiv","version":4}},"canonical_sha256":"da85379baf82f37722f534fda14aa1db76ae330a9ec2a18393ff3bb44b40a0b0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"da85379baf82f37722f534fda14aa1db76ae330a9ec2a18393ff3bb44b40a0b0","first_computed_at":"2026-05-18T03:26:51.996689Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:26:51.996689Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9RSR1LkY7Zv/q6M45r5ZlJt8Rhp4vYbXdJ2nZO/jL88KP42KiK1pTbVqaONKovWOrZ5upMjNiRl9wQMtPdC2AQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:26:51.997349Z","signed_message":"canonical_sha256_bytes"},"source_id":"1304.5010","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6d785c943cef6654d382df7ee10078f3373af6c062fc031f29dc3daff4d4ba34","sha256:77499219faf2ce8883747fc20ef953b2b19648a9ed58c223e7c76b3c1776512b"],"state_sha256":"fa2ecceaefcf201790751be8373042895d2df37b258f479ab791473125b72036"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ALsa1bLuPp+VrE2Rz/Je4d9x3WqQqUDp9iBFV1sdj+S81QiglgeGHqk221M8/G+WSsQxG3udVbPgj8ITyO/+DA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-20T15:19:07.885186Z","bundle_sha256":"70a621a0c5aa9bb647431e34b01d88bd6787e2a80e7332b719c1f6d2bcad7d1d"}}