{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:FPA32OAPG65DHXDLRY4UGH2WIL","short_pith_number":"pith:FPA32OAP","canonical_record":{"source":{"id":"2605.17122","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-16T19:04:09Z","cross_cats_sorted":[],"title_canon_sha256":"e79745c2e4bd46ee4e48ea0d0ace122abc5e9e4b8c7092a5b4639a6e19cfac1d","abstract_canon_sha256":"1eff925437d391c0bc52852fc4c9ef01fa64ba598020a2125cee28d7cbc620af"},"schema_version":"1.0"},"canonical_sha256":"2bc1bd380f37ba33dc6b8e39431f5642efca9fac809ec7b6a81990d36cb2c18d","source":{"kind":"arxiv","id":"2605.17122","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.17122","created_at":"2026-05-20T00:03:41Z"},{"alias_kind":"arxiv_version","alias_value":"2605.17122v1","created_at":"2026-05-20T00:03:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.17122","created_at":"2026-05-20T00:03:41Z"},{"alias_kind":"pith_short_12","alias_value":"FPA32OAPG65D","created_at":"2026-05-20T00:03:41Z"},{"alias_kind":"pith_short_16","alias_value":"FPA32OAPG65DHXDL","created_at":"2026-05-20T00:03:41Z"},{"alias_kind":"pith_short_8","alias_value":"FPA32OAP","created_at":"2026-05-20T00:03:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:FPA32OAPG65DHXDLRY4UGH2WIL","target":"record","payload":{"canonical_record":{"source":{"id":"2605.17122","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-16T19:04:09Z","cross_cats_sorted":[],"title_canon_sha256":"e79745c2e4bd46ee4e48ea0d0ace122abc5e9e4b8c7092a5b4639a6e19cfac1d","abstract_canon_sha256":"1eff925437d391c0bc52852fc4c9ef01fa64ba598020a2125cee28d7cbc620af"},"schema_version":"1.0"},"canonical_sha256":"2bc1bd380f37ba33dc6b8e39431f5642efca9fac809ec7b6a81990d36cb2c18d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T00:03:41.048003Z","signature_b64":"S0IfbTP7TnutkTY3seEwI5qS3yyCIBUKqzBJNO20BEA01fPnzxvDxr9IKOw701fHuO3mMwBsLgOlITrYcDypDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2bc1bd380f37ba33dc6b8e39431f5642efca9fac809ec7b6a81990d36cb2c18d","last_reissued_at":"2026-05-20T00:03:41.047188Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T00:03:41.047188Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.17122","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":"2026-05-20T00:03:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5dNE4fUQjyHi0scYuyeFdarigUbrFmbjp84dKFBxjOjmvJgDnYDEiJURsXmnlO9fJcvLhbru5wMNuQQNGyvvAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T05:51:57.604016Z"},"content_sha256":"3793b08a0aca9c27ab3a501385a936b54211ae2884d32e83079f6516131ccdb7","schema_version":"1.0","event_id":"sha256:3793b08a0aca9c27ab3a501385a936b54211ae2884d32e83079f6516131ccdb7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:FPA32OAPG65DHXDLRY4UGH2WIL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Codes and designs in multivariate $Q$-polynomial association schemes","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jing Wang, Minjia Shi, Patrick Sol\\'e","submitted_at":"2026-05-16T19:04:09Z","abstract_excerpt":"We generalize the fundamental bounds of Delsarte thesis (1973) on codes of given degree and designs of given strength in the new setting of Bannai et al. (2025). We assume the scheme is weakly metric in the sense of (Sol\\'e, 1989). We give upper bounds on the size of codes of given degree, and also on the size of codes with a given number of pairwise distances. Codes meeting these bounds are characterized by the identification\n  of suitable annihilators with the degree (resp. distance) Wilson polynomial. We give two analogues of the Rao bound on the size of designs with given strength. Designs"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.17122","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.17122/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-19T22:33:23.782780Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T22:01:58.031769Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"b7e41db74b5c6997528105fbd70f833cb4cac86ed15c82b658c4ede796afa9d8"},"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":"2026-05-20T00:03:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2vJOlUEI9Eg4WpV59hUS/CbV121KvqVYf8FV3A5MNr16Ru+i+guNVY3XwM6uoWN9OCwtR1/215+d9VeiuOI0AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T05:51:57.604471Z"},"content_sha256":"6712430ecd7d53c4ddd2abae4c8e98caa2478e89687339ce9651005fd93848a2","schema_version":"1.0","event_id":"sha256:6712430ecd7d53c4ddd2abae4c8e98caa2478e89687339ce9651005fd93848a2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FPA32OAPG65DHXDLRY4UGH2WIL/bundle.json","state_url":"https://pith.science/pith/FPA32OAPG65DHXDLRY4UGH2WIL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FPA32OAPG65DHXDLRY4UGH2WIL/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":"2026-05-30T05:51:57Z","links":{"resolver":"https://pith.science/pith/FPA32OAPG65DHXDLRY4UGH2WIL","bundle":"https://pith.science/pith/FPA32OAPG65DHXDLRY4UGH2WIL/bundle.json","state":"https://pith.science/pith/FPA32OAPG65DHXDLRY4UGH2WIL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FPA32OAPG65DHXDLRY4UGH2WIL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:FPA32OAPG65DHXDLRY4UGH2WIL","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":"1eff925437d391c0bc52852fc4c9ef01fa64ba598020a2125cee28d7cbc620af","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-16T19:04:09Z","title_canon_sha256":"e79745c2e4bd46ee4e48ea0d0ace122abc5e9e4b8c7092a5b4639a6e19cfac1d"},"schema_version":"1.0","source":{"id":"2605.17122","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.17122","created_at":"2026-05-20T00:03:41Z"},{"alias_kind":"arxiv_version","alias_value":"2605.17122v1","created_at":"2026-05-20T00:03:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.17122","created_at":"2026-05-20T00:03:41Z"},{"alias_kind":"pith_short_12","alias_value":"FPA32OAPG65D","created_at":"2026-05-20T00:03:41Z"},{"alias_kind":"pith_short_16","alias_value":"FPA32OAPG65DHXDL","created_at":"2026-05-20T00:03:41Z"},{"alias_kind":"pith_short_8","alias_value":"FPA32OAP","created_at":"2026-05-20T00:03:41Z"}],"graph_snapshots":[{"event_id":"sha256:6712430ecd7d53c4ddd2abae4c8e98caa2478e89687339ce9651005fd93848a2","target":"graph","created_at":"2026-05-20T00:03:41Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[{"findings_count":0,"name":"ai_meta_artifact","ran_at":"2026-05-19T22:33:23.782780Z","status":"skipped","version":"1.0.0"},{"findings_count":0,"name":"claim_evidence","ran_at":"2026-05-19T22:01:58.031769Z","status":"completed","version":"1.0.0"}],"endpoint":"/pith/2605.17122/integrity.json","findings":[],"snapshot_sha256":"b7e41db74b5c6997528105fbd70f833cb4cac86ed15c82b658c4ede796afa9d8","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We generalize the fundamental bounds of Delsarte thesis (1973) on codes of given degree and designs of given strength in the new setting of Bannai et al. (2025). We assume the scheme is weakly metric in the sense of (Sol\\'e, 1989). We give upper bounds on the size of codes of given degree, and also on the size of codes with a given number of pairwise distances. Codes meeting these bounds are characterized by the identification\n  of suitable annihilators with the degree (resp. distance) Wilson polynomial. We give two analogues of the Rao bound on the size of designs with given strength. Designs","authors_text":"Jing Wang, Minjia Shi, Patrick Sol\\'e","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-16T19:04:09Z","title":"Codes and designs in multivariate $Q$-polynomial association schemes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.17122","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":"sha256:3793b08a0aca9c27ab3a501385a936b54211ae2884d32e83079f6516131ccdb7","target":"record","created_at":"2026-05-20T00:03:41Z","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":"1eff925437d391c0bc52852fc4c9ef01fa64ba598020a2125cee28d7cbc620af","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-16T19:04:09Z","title_canon_sha256":"e79745c2e4bd46ee4e48ea0d0ace122abc5e9e4b8c7092a5b4639a6e19cfac1d"},"schema_version":"1.0","source":{"id":"2605.17122","kind":"arxiv","version":1}},"canonical_sha256":"2bc1bd380f37ba33dc6b8e39431f5642efca9fac809ec7b6a81990d36cb2c18d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2bc1bd380f37ba33dc6b8e39431f5642efca9fac809ec7b6a81990d36cb2c18d","first_computed_at":"2026-05-20T00:03:41.047188Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-20T00:03:41.047188Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"S0IfbTP7TnutkTY3seEwI5qS3yyCIBUKqzBJNO20BEA01fPnzxvDxr9IKOw701fHuO3mMwBsLgOlITrYcDypDw==","signature_status":"signed_v1","signed_at":"2026-05-20T00:03:41.048003Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.17122","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3793b08a0aca9c27ab3a501385a936b54211ae2884d32e83079f6516131ccdb7","sha256:6712430ecd7d53c4ddd2abae4c8e98caa2478e89687339ce9651005fd93848a2"],"state_sha256":"6665fe7cf283f7b5faaa5923b726c49da6b8e26e8978b44af7fa9415ba5d4cc7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sqfrgY8JuN4rie2Tf9ryX4bo2reCC2OFYEXblgTU9Ao3Ty4y8ruv9mpATnmZ/ZHwKJRAeZrkD/UCG9VDNvm6Dg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T05:51:57.607058Z","bundle_sha256":"88f6e4a3ca406a76bb6540ae3c8134d8e2431184ac98f180afe3675914d29e00"}}