{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:HKZJ3COTCFY6RIUGRWNYJ5ZG6D","short_pith_number":"pith:HKZJ3COT","canonical_record":{"source":{"id":"2606.28611","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-26T21:11:11Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"ea61cd0b91193c08f0be0fd7ec7dddd742494e0d496ed7796e24a6520640ea9b","abstract_canon_sha256":"91a0f07256a3a8112ccd1a61fcf04cdcc53af693a8906b181831d677c6e8579c"},"schema_version":"1.0"},"canonical_sha256":"3ab29d89d31171e8a2868d9b84f726f0e987ecc3d771f0c3f09d85dee4a969ea","source":{"kind":"arxiv","id":"2606.28611","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.28611","created_at":"2026-06-30T00:15:20Z"},{"alias_kind":"arxiv_version","alias_value":"2606.28611v1","created_at":"2026-06-30T00:15:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.28611","created_at":"2026-06-30T00:15:20Z"},{"alias_kind":"pith_short_12","alias_value":"HKZJ3COTCFY6","created_at":"2026-06-30T00:15:20Z"},{"alias_kind":"pith_short_16","alias_value":"HKZJ3COTCFY6RIUG","created_at":"2026-06-30T00:15:20Z"},{"alias_kind":"pith_short_8","alias_value":"HKZJ3COT","created_at":"2026-06-30T00:15:20Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:HKZJ3COTCFY6RIUGRWNYJ5ZG6D","target":"record","payload":{"canonical_record":{"source":{"id":"2606.28611","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-26T21:11:11Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"ea61cd0b91193c08f0be0fd7ec7dddd742494e0d496ed7796e24a6520640ea9b","abstract_canon_sha256":"91a0f07256a3a8112ccd1a61fcf04cdcc53af693a8906b181831d677c6e8579c"},"schema_version":"1.0"},"canonical_sha256":"3ab29d89d31171e8a2868d9b84f726f0e987ecc3d771f0c3f09d85dee4a969ea","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-30T00:15:20.547990Z","signature_b64":"12fTLZM25zCUL2t6IIRatkQI3Zj7IomM5YzHnB5ZFPRxe1IrCnx7zNsfqCOZnqcoLEeD6fDVqDzGkxlF8CTjAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3ab29d89d31171e8a2868d9b84f726f0e987ecc3d771f0c3f09d85dee4a969ea","last_reissued_at":"2026-06-30T00:15:20.547598Z","signature_status":"signed_v1","first_computed_at":"2026-06-30T00:15:20.547598Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.28611","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":"2026-06-30T00:15:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+l3HXk0RlLSwTKqdALsmPZeM+PXj4jXHq6AsUituoPz2X4wMfb9dTRA6CiQauUhA4xnrJ+VCx5bovyo5YyYiDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T16:34:09.449301Z"},"content_sha256":"298e88dcd9d9049befa9b59d06238cd053859bf4d2c9325937b916feb7f59a19","schema_version":"1.0","event_id":"sha256:298e88dcd9d9049befa9b59d06238cd053859bf4d2c9325937b916feb7f59a19"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:HKZJ3COTCFY6RIUGRWNYJ5ZG6D","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Structure of Cayley Codes","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.CO","authors_text":"Cheryl E. Praeger, Daniel Rademacher, Vishnuram Arumugam","submitted_at":"2026-06-26T21:11:11Z","abstract_excerpt":"Cayley codes, introduced by Kaufman and Wigderson, are linear codes constructed from a Cayley graph and a smaller linear code. We explore general properties of the class of Cayley codes for finite groups. In particular we give a reduction to Cayley codes for connected Cayley graphs that maintains code properties such as rate, minimum distance and symmetry. Also, for a given Cayley code, we identify a family of symmetric Cayley codes, each associated with a normal edge-transitive Cayley graph, such that the given Cayley code embeds into the direct sum of the symmetric Cayley codes. We analyse s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.28611","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/2606.28611/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-06-30T00:15:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"d5H6Nt4iGBMnJMET/N9HVE20qUHwA7UzASiAp30hJYfOJxlYEqcdX+t0vZTyxHq7VM4nmEePphAliOInIIIQAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T16:34:09.449692Z"},"content_sha256":"bba74cd5da57a5e32aa12f68b7466469fcbca094a7766bfad578d6d773d38274","schema_version":"1.0","event_id":"sha256:bba74cd5da57a5e32aa12f68b7466469fcbca094a7766bfad578d6d773d38274"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HKZJ3COTCFY6RIUGRWNYJ5ZG6D/bundle.json","state_url":"https://pith.science/pith/HKZJ3COTCFY6RIUGRWNYJ5ZG6D/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HKZJ3COTCFY6RIUGRWNYJ5ZG6D/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":"2026-07-01T16:34:09Z","links":{"resolver":"https://pith.science/pith/HKZJ3COTCFY6RIUGRWNYJ5ZG6D","bundle":"https://pith.science/pith/HKZJ3COTCFY6RIUGRWNYJ5ZG6D/bundle.json","state":"https://pith.science/pith/HKZJ3COTCFY6RIUGRWNYJ5ZG6D/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HKZJ3COTCFY6RIUGRWNYJ5ZG6D/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:HKZJ3COTCFY6RIUGRWNYJ5ZG6D","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":"91a0f07256a3a8112ccd1a61fcf04cdcc53af693a8906b181831d677c6e8579c","cross_cats_sorted":["math.GR"],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-26T21:11:11Z","title_canon_sha256":"ea61cd0b91193c08f0be0fd7ec7dddd742494e0d496ed7796e24a6520640ea9b"},"schema_version":"1.0","source":{"id":"2606.28611","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.28611","created_at":"2026-06-30T00:15:20Z"},{"alias_kind":"arxiv_version","alias_value":"2606.28611v1","created_at":"2026-06-30T00:15:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.28611","created_at":"2026-06-30T00:15:20Z"},{"alias_kind":"pith_short_12","alias_value":"HKZJ3COTCFY6","created_at":"2026-06-30T00:15:20Z"},{"alias_kind":"pith_short_16","alias_value":"HKZJ3COTCFY6RIUG","created_at":"2026-06-30T00:15:20Z"},{"alias_kind":"pith_short_8","alias_value":"HKZJ3COT","created_at":"2026-06-30T00:15:20Z"}],"graph_snapshots":[{"event_id":"sha256:bba74cd5da57a5e32aa12f68b7466469fcbca094a7766bfad578d6d773d38274","target":"graph","created_at":"2026-06-30T00:15:20Z","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":[],"endpoint":"/pith/2606.28611/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Cayley codes, introduced by Kaufman and Wigderson, are linear codes constructed from a Cayley graph and a smaller linear code. We explore general properties of the class of Cayley codes for finite groups. In particular we give a reduction to Cayley codes for connected Cayley graphs that maintains code properties such as rate, minimum distance and symmetry. Also, for a given Cayley code, we identify a family of symmetric Cayley codes, each associated with a normal edge-transitive Cayley graph, such that the given Cayley code embeds into the direct sum of the symmetric Cayley codes. We analyse s","authors_text":"Cheryl E. Praeger, Daniel Rademacher, Vishnuram Arumugam","cross_cats":["math.GR"],"headline":"","license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-26T21:11:11Z","title":"Structure of Cayley Codes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.28611","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":"sha256:298e88dcd9d9049befa9b59d06238cd053859bf4d2c9325937b916feb7f59a19","target":"record","created_at":"2026-06-30T00:15:20Z","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":"91a0f07256a3a8112ccd1a61fcf04cdcc53af693a8906b181831d677c6e8579c","cross_cats_sorted":["math.GR"],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-26T21:11:11Z","title_canon_sha256":"ea61cd0b91193c08f0be0fd7ec7dddd742494e0d496ed7796e24a6520640ea9b"},"schema_version":"1.0","source":{"id":"2606.28611","kind":"arxiv","version":1}},"canonical_sha256":"3ab29d89d31171e8a2868d9b84f726f0e987ecc3d771f0c3f09d85dee4a969ea","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3ab29d89d31171e8a2868d9b84f726f0e987ecc3d771f0c3f09d85dee4a969ea","first_computed_at":"2026-06-30T00:15:20.547598Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-30T00:15:20.547598Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"12fTLZM25zCUL2t6IIRatkQI3Zj7IomM5YzHnB5ZFPRxe1IrCnx7zNsfqCOZnqcoLEeD6fDVqDzGkxlF8CTjAA==","signature_status":"signed_v1","signed_at":"2026-06-30T00:15:20.547990Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.28611","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:298e88dcd9d9049befa9b59d06238cd053859bf4d2c9325937b916feb7f59a19","sha256:bba74cd5da57a5e32aa12f68b7466469fcbca094a7766bfad578d6d773d38274"],"state_sha256":"373f62ea051ec6f2a9cc5326752db8fbe3c3b7b312ff4d722467191a86fd37fb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FWNHOm1EAU2P9/06zfTguchpjTYLBmKII5KwCdIRWOivCaBjxZ9BfnOblSvc5D0/jEriRiCFZKPzVZFm6GSyCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-01T16:34:09.451740Z","bundle_sha256":"939b94a915df8588343657d71fff4fad1993ff2f908c49991177298439d766c7"}}