{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:BKEF6ATXJSSMSVHYMVX5Z7FZNX","short_pith_number":"pith:BKEF6ATX","canonical_record":{"source":{"id":"1104.4692","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-04-25T07:19:57Z","cross_cats_sorted":[],"title_canon_sha256":"f2317e8932f874cb6bb76abf9fd5dc278fb9082064a469ab0cb147087ae6488d","abstract_canon_sha256":"1d64adcfd6ef1e7061f6b4e7c2e386e20ad1ffb4f4c818c6a56815a99d669877"},"schema_version":"1.0"},"canonical_sha256":"0a885f02774ca4c954f8656fdcfcb96de9597b4c5322f6894ddf10fd095c39b9","source":{"kind":"arxiv","id":"1104.4692","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1104.4692","created_at":"2026-05-18T04:23:33Z"},{"alias_kind":"arxiv_version","alias_value":"1104.4692v1","created_at":"2026-05-18T04:23:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1104.4692","created_at":"2026-05-18T04:23:33Z"},{"alias_kind":"pith_short_12","alias_value":"BKEF6ATXJSSM","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_16","alias_value":"BKEF6ATXJSSMSVHY","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_8","alias_value":"BKEF6ATX","created_at":"2026-05-18T12:26:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:BKEF6ATXJSSMSVHYMVX5Z7FZNX","target":"record","payload":{"canonical_record":{"source":{"id":"1104.4692","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-04-25T07:19:57Z","cross_cats_sorted":[],"title_canon_sha256":"f2317e8932f874cb6bb76abf9fd5dc278fb9082064a469ab0cb147087ae6488d","abstract_canon_sha256":"1d64adcfd6ef1e7061f6b4e7c2e386e20ad1ffb4f4c818c6a56815a99d669877"},"schema_version":"1.0"},"canonical_sha256":"0a885f02774ca4c954f8656fdcfcb96de9597b4c5322f6894ddf10fd095c39b9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:23:33.612672Z","signature_b64":"j9EYvtf6cvySxzw5vaLGmoemgZb+5RJ+sRKeJZKZqzPmqvN7UNujUkZ+zNAvNzbdZn6O6IhIRlyFLwKETpixAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0a885f02774ca4c954f8656fdcfcb96de9597b4c5322f6894ddf10fd095c39b9","last_reissued_at":"2026-05-18T04:23:33.612162Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:23:33.612162Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1104.4692","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":"2026-05-18T04:23:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PHUZ53O18sYRFBURn9YybtcXshG/i83WRum1zsGlBoKJbnNNovqnrImnffJWF5W95DlPTkccJtE9ItjFT9ZoBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T00:34:58.959930Z"},"content_sha256":"d83fe2c4fdc0855f4d6649b88d47437677296a4d2705b42a993054f48142b5d5","schema_version":"1.0","event_id":"sha256:d83fe2c4fdc0855f4d6649b88d47437677296a4d2705b42a993054f48142b5d5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:BKEF6ATXJSSMSVHYMVX5Z7FZNX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Complex spherical designs and codes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Aidan Roy, Sho Suda","submitted_at":"2011-04-25T07:19:57Z","abstract_excerpt":"Real spherical designs and real and complex projective designs have been shown by Delsarte, Goethals, and Seidel to give rise to association schemes when the strength of the design is high compared to its degree as a code. In contrast, designs on the complex unit sphere remain relatively uninvestigated, despite their importance in numerous applications. In this paper we develop the notion of a complex spherical design and show how many such designs carry the structure of an association scheme. In contrast with the real spherical designs and the real and complex projective designs, these associ"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.4692","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":"2026-05-18T04:23:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RY1W3nZgmKH50Y1Ixbp0R6ACfc2BFWcgDPZ03c4Jnt59D2fGytAINk/pjejGQYebJN4pag3WQG9bUfnW+J/QBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T00:34:58.960659Z"},"content_sha256":"b58462291da1609a1585e3313d34d62203c5b0ea123c6a0238dcedfee8169827","schema_version":"1.0","event_id":"sha256:b58462291da1609a1585e3313d34d62203c5b0ea123c6a0238dcedfee8169827"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BKEF6ATXJSSMSVHYMVX5Z7FZNX/bundle.json","state_url":"https://pith.science/pith/BKEF6ATXJSSMSVHYMVX5Z7FZNX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BKEF6ATXJSSMSVHYMVX5Z7FZNX/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":"2026-05-26T00:34:58Z","links":{"resolver":"https://pith.science/pith/BKEF6ATXJSSMSVHYMVX5Z7FZNX","bundle":"https://pith.science/pith/BKEF6ATXJSSMSVHYMVX5Z7FZNX/bundle.json","state":"https://pith.science/pith/BKEF6ATXJSSMSVHYMVX5Z7FZNX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BKEF6ATXJSSMSVHYMVX5Z7FZNX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:BKEF6ATXJSSMSVHYMVX5Z7FZNX","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":"1d64adcfd6ef1e7061f6b4e7c2e386e20ad1ffb4f4c818c6a56815a99d669877","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-04-25T07:19:57Z","title_canon_sha256":"f2317e8932f874cb6bb76abf9fd5dc278fb9082064a469ab0cb147087ae6488d"},"schema_version":"1.0","source":{"id":"1104.4692","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1104.4692","created_at":"2026-05-18T04:23:33Z"},{"alias_kind":"arxiv_version","alias_value":"1104.4692v1","created_at":"2026-05-18T04:23:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1104.4692","created_at":"2026-05-18T04:23:33Z"},{"alias_kind":"pith_short_12","alias_value":"BKEF6ATXJSSM","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_16","alias_value":"BKEF6ATXJSSMSVHY","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_8","alias_value":"BKEF6ATX","created_at":"2026-05-18T12:26:24Z"}],"graph_snapshots":[{"event_id":"sha256:b58462291da1609a1585e3313d34d62203c5b0ea123c6a0238dcedfee8169827","target":"graph","created_at":"2026-05-18T04:23:33Z","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":"Real spherical designs and real and complex projective designs have been shown by Delsarte, Goethals, and Seidel to give rise to association schemes when the strength of the design is high compared to its degree as a code. In contrast, designs on the complex unit sphere remain relatively uninvestigated, despite their importance in numerous applications. In this paper we develop the notion of a complex spherical design and show how many such designs carry the structure of an association scheme. In contrast with the real spherical designs and the real and complex projective designs, these associ","authors_text":"Aidan Roy, Sho Suda","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-04-25T07:19:57Z","title":"Complex spherical designs and codes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.4692","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:d83fe2c4fdc0855f4d6649b88d47437677296a4d2705b42a993054f48142b5d5","target":"record","created_at":"2026-05-18T04:23:33Z","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":"1d64adcfd6ef1e7061f6b4e7c2e386e20ad1ffb4f4c818c6a56815a99d669877","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-04-25T07:19:57Z","title_canon_sha256":"f2317e8932f874cb6bb76abf9fd5dc278fb9082064a469ab0cb147087ae6488d"},"schema_version":"1.0","source":{"id":"1104.4692","kind":"arxiv","version":1}},"canonical_sha256":"0a885f02774ca4c954f8656fdcfcb96de9597b4c5322f6894ddf10fd095c39b9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0a885f02774ca4c954f8656fdcfcb96de9597b4c5322f6894ddf10fd095c39b9","first_computed_at":"2026-05-18T04:23:33.612162Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:23:33.612162Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"j9EYvtf6cvySxzw5vaLGmoemgZb+5RJ+sRKeJZKZqzPmqvN7UNujUkZ+zNAvNzbdZn6O6IhIRlyFLwKETpixAg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:23:33.612672Z","signed_message":"canonical_sha256_bytes"},"source_id":"1104.4692","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d83fe2c4fdc0855f4d6649b88d47437677296a4d2705b42a993054f48142b5d5","sha256:b58462291da1609a1585e3313d34d62203c5b0ea123c6a0238dcedfee8169827"],"state_sha256":"493f627f8c25e4a003ac67ce6186100c8f04372a4f058edbdcf36db2322f59c6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/dJTqdrt2KCqaM+j7mnaN5w6ToA3ClLbJGZF7agjohp98Nd6uLSSkUPij/dtlVHUUXTZjAeyALFnNFpvV8RMCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T00:34:58.965860Z","bundle_sha256":"2c1f749bf2dd8c45dc321ff88b54c2753c034a93cc1ddc56a18db65b37ea4f19"}}