{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:KCSL4SOLPM2LX6SDAOTNOYZBEC","short_pith_number":"pith:KCSL4SOL","canonical_record":{"source":{"id":"1510.03406","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2015-10-12T19:38:28Z","cross_cats_sorted":[],"title_canon_sha256":"fe1aea776bdf1a010c7373eeafdea5dd5b2ff510357ff8630f7e967b93519e00","abstract_canon_sha256":"ada7fa5cf0387d0508cc1c095de58fe79d53cedf993d6b13d3baff1ff6484359"},"schema_version":"1.0"},"canonical_sha256":"50a4be49cb7b34bbfa4303a6d7632120be60ff66cd8add371ac7e8585f9f568f","source":{"kind":"arxiv","id":"1510.03406","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.03406","created_at":"2026-05-18T01:02:13Z"},{"alias_kind":"arxiv_version","alias_value":"1510.03406v2","created_at":"2026-05-18T01:02:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.03406","created_at":"2026-05-18T01:02:13Z"},{"alias_kind":"pith_short_12","alias_value":"KCSL4SOLPM2L","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"KCSL4SOLPM2LX6SD","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"KCSL4SOL","created_at":"2026-05-18T12:29:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:KCSL4SOLPM2LX6SDAOTNOYZBEC","target":"record","payload":{"canonical_record":{"source":{"id":"1510.03406","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2015-10-12T19:38:28Z","cross_cats_sorted":[],"title_canon_sha256":"fe1aea776bdf1a010c7373eeafdea5dd5b2ff510357ff8630f7e967b93519e00","abstract_canon_sha256":"ada7fa5cf0387d0508cc1c095de58fe79d53cedf993d6b13d3baff1ff6484359"},"schema_version":"1.0"},"canonical_sha256":"50a4be49cb7b34bbfa4303a6d7632120be60ff66cd8add371ac7e8585f9f568f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:02:13.608028Z","signature_b64":"Hkuf2Su2i3RKWbwdGU4zNGulvcAQ1a3WcGLo5z6Y9vgx4z4oCFWNMtUTIsWifDrvzVZD2NKW0LTFqNW4D182Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"50a4be49cb7b34bbfa4303a6d7632120be60ff66cd8add371ac7e8585f9f568f","last_reissued_at":"2026-05-18T01:02:13.607425Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:02:13.607425Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1510.03406","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":"2026-05-18T01:02:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"R19+ZrEH2rU3eKAWOT58b37OZVDazEpD4ILwiOCyoWnvwoqdFyG2QsNRPVsEP9ZM7TK4sQY2qiIsZYt7RCotDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T08:04:13.088339Z"},"content_sha256":"b27a09a8ae0be5b66f8f2f717c0932b5c4bf2ba3d5069e3f642e9f4c543d6514","schema_version":"1.0","event_id":"sha256:b27a09a8ae0be5b66f8f2f717c0932b5c4bf2ba3d5069e3f642e9f4c543d6514"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:KCSL4SOLPM2LX6SDAOTNOYZBEC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Energy bounds for codes and designs in Hamming spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Douglas P. Hardin, Edward B. Saff, Maya M. Stoyanova, Peter D. Dragnev, Peter G. Boyvalenkov","submitted_at":"2015-10-12T19:38:28Z","abstract_excerpt":"We obtain universal bounds on the energy of codes and for designs in Hamming spaces. Our bounds hold for a large class of potential functions, allow unified treatment, and can be viewed as a generalization of the Levenshtein bounds for maximal codes."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.03406","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":"2026-05-18T01:02:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fI3otaopQlHGBTsK+OsrCZBflUvnhEICblVB5P3Hz/u99JBy9HIeby3rWxJDx30R+y0qWLFC95xneoq/D5ngDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T08:04:13.088679Z"},"content_sha256":"719e9e9847c6904e92e55d7c79bf4cd2208608425d038ff673e0dacacc02179d","schema_version":"1.0","event_id":"sha256:719e9e9847c6904e92e55d7c79bf4cd2208608425d038ff673e0dacacc02179d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KCSL4SOLPM2LX6SDAOTNOYZBEC/bundle.json","state_url":"https://pith.science/pith/KCSL4SOLPM2LX6SDAOTNOYZBEC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KCSL4SOLPM2LX6SDAOTNOYZBEC/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":"2026-06-02T08:04:13Z","links":{"resolver":"https://pith.science/pith/KCSL4SOLPM2LX6SDAOTNOYZBEC","bundle":"https://pith.science/pith/KCSL4SOLPM2LX6SDAOTNOYZBEC/bundle.json","state":"https://pith.science/pith/KCSL4SOLPM2LX6SDAOTNOYZBEC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KCSL4SOLPM2LX6SDAOTNOYZBEC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:KCSL4SOLPM2LX6SDAOTNOYZBEC","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":"ada7fa5cf0387d0508cc1c095de58fe79d53cedf993d6b13d3baff1ff6484359","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2015-10-12T19:38:28Z","title_canon_sha256":"fe1aea776bdf1a010c7373eeafdea5dd5b2ff510357ff8630f7e967b93519e00"},"schema_version":"1.0","source":{"id":"1510.03406","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.03406","created_at":"2026-05-18T01:02:13Z"},{"alias_kind":"arxiv_version","alias_value":"1510.03406v2","created_at":"2026-05-18T01:02:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.03406","created_at":"2026-05-18T01:02:13Z"},{"alias_kind":"pith_short_12","alias_value":"KCSL4SOLPM2L","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"KCSL4SOLPM2LX6SD","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"KCSL4SOL","created_at":"2026-05-18T12:29:27Z"}],"graph_snapshots":[{"event_id":"sha256:719e9e9847c6904e92e55d7c79bf4cd2208608425d038ff673e0dacacc02179d","target":"graph","created_at":"2026-05-18T01:02:13Z","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 obtain universal bounds on the energy of codes and for designs in Hamming spaces. Our bounds hold for a large class of potential functions, allow unified treatment, and can be viewed as a generalization of the Levenshtein bounds for maximal codes.","authors_text":"Douglas P. Hardin, Edward B. Saff, Maya M. Stoyanova, Peter D. Dragnev, Peter G. Boyvalenkov","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2015-10-12T19:38:28Z","title":"Energy bounds for codes and designs in Hamming spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.03406","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:b27a09a8ae0be5b66f8f2f717c0932b5c4bf2ba3d5069e3f642e9f4c543d6514","target":"record","created_at":"2026-05-18T01:02:13Z","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":"ada7fa5cf0387d0508cc1c095de58fe79d53cedf993d6b13d3baff1ff6484359","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2015-10-12T19:38:28Z","title_canon_sha256":"fe1aea776bdf1a010c7373eeafdea5dd5b2ff510357ff8630f7e967b93519e00"},"schema_version":"1.0","source":{"id":"1510.03406","kind":"arxiv","version":2}},"canonical_sha256":"50a4be49cb7b34bbfa4303a6d7632120be60ff66cd8add371ac7e8585f9f568f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"50a4be49cb7b34bbfa4303a6d7632120be60ff66cd8add371ac7e8585f9f568f","first_computed_at":"2026-05-18T01:02:13.607425Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:02:13.607425Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Hkuf2Su2i3RKWbwdGU4zNGulvcAQ1a3WcGLo5z6Y9vgx4z4oCFWNMtUTIsWifDrvzVZD2NKW0LTFqNW4D182Aw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:02:13.608028Z","signed_message":"canonical_sha256_bytes"},"source_id":"1510.03406","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b27a09a8ae0be5b66f8f2f717c0932b5c4bf2ba3d5069e3f642e9f4c543d6514","sha256:719e9e9847c6904e92e55d7c79bf4cd2208608425d038ff673e0dacacc02179d"],"state_sha256":"be6e0bac4e7d7ee04f078cf4f5ff023d7b513f72ca7b8f8a2dd2adf0b3428340"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yyCj1be8sTHc8rY/jULQOpiscw+d04tGCioqf0lFXCKtGH9dH0Svw45LiLKldsLf++s5bqkgqhd0QMQAcokRBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T08:04:13.090623Z","bundle_sha256":"ef2b8eaf11e8a9981477487129a39634effda34fba6ed0472e4dc2e24e0a4c57"}}