{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:F3K6IW2YZDJUYHVW5BU3VAFGGZ","short_pith_number":"pith:F3K6IW2Y","canonical_record":{"source":{"id":"1511.04494","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-11-14T02:39:55Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"a3187b742ae9f2884711e40899590ef8f00d98d7704d18fd7b7ec6f102eafb7d","abstract_canon_sha256":"ccdb432a4c9d38253911f1e046925fccf577a286ea59b0be2febd7f03a2b66e7"},"schema_version":"1.0"},"canonical_sha256":"2ed5e45b58c8d34c1eb6e869ba80a636796af2430b5fc0754a7b8fc9fb380d7f","source":{"kind":"arxiv","id":"1511.04494","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.04494","created_at":"2026-05-18T01:03:36Z"},{"alias_kind":"arxiv_version","alias_value":"1511.04494v3","created_at":"2026-05-18T01:03:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.04494","created_at":"2026-05-18T01:03:36Z"},{"alias_kind":"pith_short_12","alias_value":"F3K6IW2YZDJU","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_16","alias_value":"F3K6IW2YZDJUYHVW","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_8","alias_value":"F3K6IW2Y","created_at":"2026-05-18T12:29:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:F3K6IW2YZDJUYHVW5BU3VAFGGZ","target":"record","payload":{"canonical_record":{"source":{"id":"1511.04494","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-11-14T02:39:55Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"a3187b742ae9f2884711e40899590ef8f00d98d7704d18fd7b7ec6f102eafb7d","abstract_canon_sha256":"ccdb432a4c9d38253911f1e046925fccf577a286ea59b0be2febd7f03a2b66e7"},"schema_version":"1.0"},"canonical_sha256":"2ed5e45b58c8d34c1eb6e869ba80a636796af2430b5fc0754a7b8fc9fb380d7f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:03:36.802173Z","signature_b64":"WZKg74jaGJVazewkvlvX7UvMPqLdXotdUk7nrIvDcwdujeavCf3lYqhUIRbpIBQuK7JhcIMnR3zAlNGCkj6yAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2ed5e45b58c8d34c1eb6e869ba80a636796af2430b5fc0754a7b8fc9fb380d7f","last_reissued_at":"2026-05-18T01:03:36.801447Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:03:36.801447Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1511.04494","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":"2026-05-18T01:03:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qXBJ55q0SUnf6cjgtTra2OJqgpkQH2qU7t49pmZDrzDnm2A4sAsBoJso27zgtq0zNwrwKalqhCIGN2pQNZ4FBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-24T17:20:36.398747Z"},"content_sha256":"2f60d61bc80fb81f0f6286ab6c1e72069380ca51c7d9080d445b4e0ac45c1832","schema_version":"1.0","event_id":"sha256:2f60d61bc80fb81f0f6286ab6c1e72069380ca51c7d9080d445b4e0ac45c1832"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:F3K6IW2YZDJUYHVW5BU3VAFGGZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Constructing Permutation Arrays from Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Avi Levy, I. Hal Sudborough, Sergey Bereg","submitted_at":"2015-11-14T02:39:55Z","abstract_excerpt":"Let M(n, d) be the maximum size of a permutation array on n symbols with pairwise Hamming distance at least d. We use various combinatorial, algebraic, and computational methods to improve lower bounds for M(n, d). We compute the Hamming distances of affine semilinear groups and projective semilinear groups, and unions of cosets of AGL(1,q) and PGL(2,q) with Frobenius maps to obtain new, improved lower bounds for M(n,d). We give new randomized algorithms. We give better lower bounds for M(n,d) also using new theorems concerning the contraction operation. For example, we prove a quadratic lower"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.04494","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":"2026-05-18T01:03:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lwk8Fw3hw2oYu+sLYonjZScgwUXvr/HFS89iKkyyJRtO2MaSEu8KeiDKEUtyCt7oLgMTpAMU5RbJqggj+awpDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-24T17:20:36.399391Z"},"content_sha256":"0d34c5970dda9f518ec4b0145588f30a755bc10763ba5223b030fd658c908d32","schema_version":"1.0","event_id":"sha256:0d34c5970dda9f518ec4b0145588f30a755bc10763ba5223b030fd658c908d32"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/F3K6IW2YZDJUYHVW5BU3VAFGGZ/bundle.json","state_url":"https://pith.science/pith/F3K6IW2YZDJUYHVW5BU3VAFGGZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/F3K6IW2YZDJUYHVW5BU3VAFGGZ/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":"2026-05-24T17:20:36Z","links":{"resolver":"https://pith.science/pith/F3K6IW2YZDJUYHVW5BU3VAFGGZ","bundle":"https://pith.science/pith/F3K6IW2YZDJUYHVW5BU3VAFGGZ/bundle.json","state":"https://pith.science/pith/F3K6IW2YZDJUYHVW5BU3VAFGGZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/F3K6IW2YZDJUYHVW5BU3VAFGGZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:F3K6IW2YZDJUYHVW5BU3VAFGGZ","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":"ccdb432a4c9d38253911f1e046925fccf577a286ea59b0be2febd7f03a2b66e7","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-11-14T02:39:55Z","title_canon_sha256":"a3187b742ae9f2884711e40899590ef8f00d98d7704d18fd7b7ec6f102eafb7d"},"schema_version":"1.0","source":{"id":"1511.04494","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.04494","created_at":"2026-05-18T01:03:36Z"},{"alias_kind":"arxiv_version","alias_value":"1511.04494v3","created_at":"2026-05-18T01:03:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.04494","created_at":"2026-05-18T01:03:36Z"},{"alias_kind":"pith_short_12","alias_value":"F3K6IW2YZDJU","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_16","alias_value":"F3K6IW2YZDJUYHVW","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_8","alias_value":"F3K6IW2Y","created_at":"2026-05-18T12:29:19Z"}],"graph_snapshots":[{"event_id":"sha256:0d34c5970dda9f518ec4b0145588f30a755bc10763ba5223b030fd658c908d32","target":"graph","created_at":"2026-05-18T01:03:36Z","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 M(n, d) be the maximum size of a permutation array on n symbols with pairwise Hamming distance at least d. We use various combinatorial, algebraic, and computational methods to improve lower bounds for M(n, d). We compute the Hamming distances of affine semilinear groups and projective semilinear groups, and unions of cosets of AGL(1,q) and PGL(2,q) with Frobenius maps to obtain new, improved lower bounds for M(n,d). We give new randomized algorithms. We give better lower bounds for M(n,d) also using new theorems concerning the contraction operation. For example, we prove a quadratic lower","authors_text":"Avi Levy, I. Hal Sudborough, Sergey Bereg","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-11-14T02:39:55Z","title":"Constructing Permutation Arrays from Groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.04494","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":"sha256:2f60d61bc80fb81f0f6286ab6c1e72069380ca51c7d9080d445b4e0ac45c1832","target":"record","created_at":"2026-05-18T01:03:36Z","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":"ccdb432a4c9d38253911f1e046925fccf577a286ea59b0be2febd7f03a2b66e7","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-11-14T02:39:55Z","title_canon_sha256":"a3187b742ae9f2884711e40899590ef8f00d98d7704d18fd7b7ec6f102eafb7d"},"schema_version":"1.0","source":{"id":"1511.04494","kind":"arxiv","version":3}},"canonical_sha256":"2ed5e45b58c8d34c1eb6e869ba80a636796af2430b5fc0754a7b8fc9fb380d7f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2ed5e45b58c8d34c1eb6e869ba80a636796af2430b5fc0754a7b8fc9fb380d7f","first_computed_at":"2026-05-18T01:03:36.801447Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:03:36.801447Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"WZKg74jaGJVazewkvlvX7UvMPqLdXotdUk7nrIvDcwdujeavCf3lYqhUIRbpIBQuK7JhcIMnR3zAlNGCkj6yAA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:03:36.802173Z","signed_message":"canonical_sha256_bytes"},"source_id":"1511.04494","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2f60d61bc80fb81f0f6286ab6c1e72069380ca51c7d9080d445b4e0ac45c1832","sha256:0d34c5970dda9f518ec4b0145588f30a755bc10763ba5223b030fd658c908d32"],"state_sha256":"016daa1083996a6732c692681631490e42652ab29e1522d844cbefdb0c23e297"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FZX61/ufzjIqdlTxgiXuB4Z6Oh3Nl2mpo3R4HnG6cWHvszWmAru0Xyy7e4LckSDHaSlJWzA58csVzfsumzw9DA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-24T17:20:36.402172Z","bundle_sha256":"a1b284995c858a3a465eaf516eeb427dec57e77f3a7b0a35775c8e7a1536700c"}}