{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:ML4UBTFCWR4AIKOAQBY6GGH4FN","short_pith_number":"pith:ML4UBTFC","canonical_record":{"source":{"id":"1011.1044","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-11-04T01:24:15Z","cross_cats_sorted":[],"title_canon_sha256":"76989638e2ae25ae0197d0585f41239357f9905211e1910993f36f00a8ba480d","abstract_canon_sha256":"03b9c6cac62093305f26b3b0c559170bfbe8a081dae8cc5c264d4131675dc0c6"},"schema_version":"1.0"},"canonical_sha256":"62f940cca2b4780429c08071e318fc2b66998cd9967c22e6c8cfda7c734d3f40","source":{"kind":"arxiv","id":"1011.1044","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1011.1044","created_at":"2026-05-18T04:37:09Z"},{"alias_kind":"arxiv_version","alias_value":"1011.1044v1","created_at":"2026-05-18T04:37:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.1044","created_at":"2026-05-18T04:37:09Z"},{"alias_kind":"pith_short_12","alias_value":"ML4UBTFCWR4A","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_16","alias_value":"ML4UBTFCWR4AIKOA","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_8","alias_value":"ML4UBTFC","created_at":"2026-05-18T12:26:10Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:ML4UBTFCWR4AIKOAQBY6GGH4FN","target":"record","payload":{"canonical_record":{"source":{"id":"1011.1044","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-11-04T01:24:15Z","cross_cats_sorted":[],"title_canon_sha256":"76989638e2ae25ae0197d0585f41239357f9905211e1910993f36f00a8ba480d","abstract_canon_sha256":"03b9c6cac62093305f26b3b0c559170bfbe8a081dae8cc5c264d4131675dc0c6"},"schema_version":"1.0"},"canonical_sha256":"62f940cca2b4780429c08071e318fc2b66998cd9967c22e6c8cfda7c734d3f40","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:37:09.732569Z","signature_b64":"8Nn8JGrLc9BKW7rVSK8j7bRDVe287bDh9jAZQk/WYurmjENmSOM+ColBmPMZGpPfPoYXJR5YM4rXypz2chZPBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"62f940cca2b4780429c08071e318fc2b66998cd9967c22e6c8cfda7c734d3f40","last_reissued_at":"2026-05-18T04:37:09.732047Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:37:09.732047Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1011.1044","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:37:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OaJLq7x+MMy0L4XopCwh0JM9KJ7ZuKiuxGBak83edMojqc4smBccTuPkp+H5U+k8kzJ7enKLYzkeqFqJkYkrDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-23T17:03:25.263533Z"},"content_sha256":"7c98513f75fce65d755b8df0d3da79022535bd699c5eefa820fad721d3f4ebf4","schema_version":"1.0","event_id":"sha256:7c98513f75fce65d755b8df0d3da79022535bd699c5eefa820fad721d3f4ebf4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:ML4UBTFCWR4AIKOAQBY6GGH4FN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Generalized Hamming schemes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Chris Godsil","submitted_at":"2010-11-04T01:24:15Z","abstract_excerpt":"We introduce a class of association schemes that generalizes the Hamming scheme. We derive generating functions for their eigenvalues, and use these to obtain a version of MacWilliams theorem."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.1044","kind":"arxiv","version":1},"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":"2026-05-18T04:37:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Oi1pPbrfREB/LcYN3ptkN404K54dPkwZc5xX/tnBfkBiXCkhaidM8GQANTrOUcy1A7vOYh+eqNOQp0ZeNMJkDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-23T17:03:25.264015Z"},"content_sha256":"120eb2714e3fafdd0646224942ee18242f52692c1d7cee68d619cce3539a64ab","schema_version":"1.0","event_id":"sha256:120eb2714e3fafdd0646224942ee18242f52692c1d7cee68d619cce3539a64ab"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ML4UBTFCWR4AIKOAQBY6GGH4FN/bundle.json","state_url":"https://pith.science/pith/ML4UBTFCWR4AIKOAQBY6GGH4FN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ML4UBTFCWR4AIKOAQBY6GGH4FN/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":"2026-05-23T17:03:25Z","links":{"resolver":"https://pith.science/pith/ML4UBTFCWR4AIKOAQBY6GGH4FN","bundle":"https://pith.science/pith/ML4UBTFCWR4AIKOAQBY6GGH4FN/bundle.json","state":"https://pith.science/pith/ML4UBTFCWR4AIKOAQBY6GGH4FN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ML4UBTFCWR4AIKOAQBY6GGH4FN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:ML4UBTFCWR4AIKOAQBY6GGH4FN","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":"03b9c6cac62093305f26b3b0c559170bfbe8a081dae8cc5c264d4131675dc0c6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-11-04T01:24:15Z","title_canon_sha256":"76989638e2ae25ae0197d0585f41239357f9905211e1910993f36f00a8ba480d"},"schema_version":"1.0","source":{"id":"1011.1044","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1011.1044","created_at":"2026-05-18T04:37:09Z"},{"alias_kind":"arxiv_version","alias_value":"1011.1044v1","created_at":"2026-05-18T04:37:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.1044","created_at":"2026-05-18T04:37:09Z"},{"alias_kind":"pith_short_12","alias_value":"ML4UBTFCWR4A","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_16","alias_value":"ML4UBTFCWR4AIKOA","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_8","alias_value":"ML4UBTFC","created_at":"2026-05-18T12:26:10Z"}],"graph_snapshots":[{"event_id":"sha256:120eb2714e3fafdd0646224942ee18242f52692c1d7cee68d619cce3539a64ab","target":"graph","created_at":"2026-05-18T04:37:09Z","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":"We introduce a class of association schemes that generalizes the Hamming scheme. We derive generating functions for their eigenvalues, and use these to obtain a version of MacWilliams theorem.","authors_text":"Chris Godsil","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-11-04T01:24:15Z","title":"Generalized Hamming schemes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.1044","kind":"arxiv","version":1},"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":"sha256:7c98513f75fce65d755b8df0d3da79022535bd699c5eefa820fad721d3f4ebf4","target":"record","created_at":"2026-05-18T04:37:09Z","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":"03b9c6cac62093305f26b3b0c559170bfbe8a081dae8cc5c264d4131675dc0c6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-11-04T01:24:15Z","title_canon_sha256":"76989638e2ae25ae0197d0585f41239357f9905211e1910993f36f00a8ba480d"},"schema_version":"1.0","source":{"id":"1011.1044","kind":"arxiv","version":1}},"canonical_sha256":"62f940cca2b4780429c08071e318fc2b66998cd9967c22e6c8cfda7c734d3f40","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"62f940cca2b4780429c08071e318fc2b66998cd9967c22e6c8cfda7c734d3f40","first_computed_at":"2026-05-18T04:37:09.732047Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:37:09.732047Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8Nn8JGrLc9BKW7rVSK8j7bRDVe287bDh9jAZQk/WYurmjENmSOM+ColBmPMZGpPfPoYXJR5YM4rXypz2chZPBw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:37:09.732569Z","signed_message":"canonical_sha256_bytes"},"source_id":"1011.1044","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7c98513f75fce65d755b8df0d3da79022535bd699c5eefa820fad721d3f4ebf4","sha256:120eb2714e3fafdd0646224942ee18242f52692c1d7cee68d619cce3539a64ab"],"state_sha256":"c7fb78e6c11806db561e6d4a1b3a7aa1a8b201235cd9a0849f54751bdf31ab81"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LRYfByJRoul+TN+mpvOvriupybKkkVDKJIDuOpogm4wLALcnYBRqi5pgjPokrd38XmVwtbC1jY/ZbzxhcdzGAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-23T17:03:25.267661Z","bundle_sha256":"501d752838c4e193d2e88be8b2cb6e3fd21aea29858f5ccf71b70c5b2852f137"}}