{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:D4HJW6HH6U2GBZW6TBGGZOXO7G","short_pith_number":"pith:D4HJW6HH","canonical_record":{"source":{"id":"2606.11987","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.IT","submitted_at":"2026-06-10T12:11:25Z","cross_cats_sorted":["math.CO","math.IT"],"title_canon_sha256":"dfc0a79afdc220ef30be0ed1ea33bf240efbdd4d333d617d336a6b9d2d0f26a7","abstract_canon_sha256":"1260ef5e6fe52d373fcb73b3dc507b28a987c9ea55b74055f5f7b2646e50d590"},"schema_version":"1.0"},"canonical_sha256":"1f0e9b78e7f53460e6de984c6cbaeef99884c9a79fb9a1a35aa291a888e81128","source":{"kind":"arxiv","id":"2606.11987","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.11987","created_at":"2026-06-11T01:10:41Z"},{"alias_kind":"arxiv_version","alias_value":"2606.11987v1","created_at":"2026-06-11T01:10:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.11987","created_at":"2026-06-11T01:10:41Z"},{"alias_kind":"pith_short_12","alias_value":"D4HJW6HH6U2G","created_at":"2026-06-11T01:10:41Z"},{"alias_kind":"pith_short_16","alias_value":"D4HJW6HH6U2GBZW6","created_at":"2026-06-11T01:10:41Z"},{"alias_kind":"pith_short_8","alias_value":"D4HJW6HH","created_at":"2026-06-11T01:10:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:D4HJW6HH6U2GBZW6TBGGZOXO7G","target":"record","payload":{"canonical_record":{"source":{"id":"2606.11987","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.IT","submitted_at":"2026-06-10T12:11:25Z","cross_cats_sorted":["math.CO","math.IT"],"title_canon_sha256":"dfc0a79afdc220ef30be0ed1ea33bf240efbdd4d333d617d336a6b9d2d0f26a7","abstract_canon_sha256":"1260ef5e6fe52d373fcb73b3dc507b28a987c9ea55b74055f5f7b2646e50d590"},"schema_version":"1.0"},"canonical_sha256":"1f0e9b78e7f53460e6de984c6cbaeef99884c9a79fb9a1a35aa291a888e81128","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-11T01:10:41.190634Z","signature_b64":"Sasr3OZxzlYHNiy10HKAryJ1iaROveC0kA0oGnZA5P8A29nvKluUPojXz4Z3N+LSpeysSVPRhnIxBORnouLVDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1f0e9b78e7f53460e6de984c6cbaeef99884c9a79fb9a1a35aa291a888e81128","last_reissued_at":"2026-06-11T01:10:41.189761Z","signature_status":"signed_v1","first_computed_at":"2026-06-11T01:10:41.189761Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.11987","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":"2026-06-11T01:10:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ODqM8xwaGx/Z/WVagwSvmqWCaVc5yQGjLIe5I6gR9ymZBvj1B4kzBnmKqHBoqFC/guptcGStvXud9lhmNl3cCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T19:23:45.398194Z"},"content_sha256":"fb391c56a45d36c9a405d876e0c7b016253783d9046856b8afafa15fd2db6a62","schema_version":"1.0","event_id":"sha256:fb391c56a45d36c9a405d876e0c7b016253783d9046856b8afafa15fd2db6a62"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:D4HJW6HH6U2GBZW6TBGGZOXO7G","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Graphical Analysis of Lifted Product Code Constructions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO","math.IT"],"primary_cat":"cs.IT","authors_text":"Kirsten D. Morris, Patricija \\v{S}apokait\\.e, Ragnar Freij-Hollanti","submitted_at":"2026-06-10T12:11:25Z","abstract_excerpt":"Lifted product codes are an important family of quantum low-density parity-check (QLDPC) codes, as they were the first QLDPC code family shown to be asymptotically good. Understanding the structure of their parity-check matrices $H_{\\mathsf{X}}$ and $H_{\\mathsf{Z}}$, as well as the associated Tanner graphs, is essential for analyzing their decoding behavior and error-floor performance. In this work, we show that the Tanner graphs of $H_{\\mathsf{X}}$ and $H_{\\mathsf{Z}}$ are indeed isomorphic, and investigate their graph-theoretical structure. We establish conditions ensuring the connectivity o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.11987","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/2606.11987/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":"2026-06-11T01:10:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fmOAfpLPJnxC3QZoj+Ct0mAo3FAv7C6/8AOigh+8NBpKsxeWC+xsJZomuotQrNvdMjXU9xv0Jsh3uCnYHcsuCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T19:23:45.399445Z"},"content_sha256":"a96909cadcfad39b69b5566fd0ff897538ffd3de48a4c765d0a14248f2d9e86e","schema_version":"1.0","event_id":"sha256:a96909cadcfad39b69b5566fd0ff897538ffd3de48a4c765d0a14248f2d9e86e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/D4HJW6HH6U2GBZW6TBGGZOXO7G/bundle.json","state_url":"https://pith.science/pith/D4HJW6HH6U2GBZW6TBGGZOXO7G/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/D4HJW6HH6U2GBZW6TBGGZOXO7G/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":"2026-06-23T19:23:45Z","links":{"resolver":"https://pith.science/pith/D4HJW6HH6U2GBZW6TBGGZOXO7G","bundle":"https://pith.science/pith/D4HJW6HH6U2GBZW6TBGGZOXO7G/bundle.json","state":"https://pith.science/pith/D4HJW6HH6U2GBZW6TBGGZOXO7G/state.json","well_known_bundle":"https://pith.science/.well-known/pith/D4HJW6HH6U2GBZW6TBGGZOXO7G/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:D4HJW6HH6U2GBZW6TBGGZOXO7G","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":"1260ef5e6fe52d373fcb73b3dc507b28a987c9ea55b74055f5f7b2646e50d590","cross_cats_sorted":["math.CO","math.IT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.IT","submitted_at":"2026-06-10T12:11:25Z","title_canon_sha256":"dfc0a79afdc220ef30be0ed1ea33bf240efbdd4d333d617d336a6b9d2d0f26a7"},"schema_version":"1.0","source":{"id":"2606.11987","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.11987","created_at":"2026-06-11T01:10:41Z"},{"alias_kind":"arxiv_version","alias_value":"2606.11987v1","created_at":"2026-06-11T01:10:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.11987","created_at":"2026-06-11T01:10:41Z"},{"alias_kind":"pith_short_12","alias_value":"D4HJW6HH6U2G","created_at":"2026-06-11T01:10:41Z"},{"alias_kind":"pith_short_16","alias_value":"D4HJW6HH6U2GBZW6","created_at":"2026-06-11T01:10:41Z"},{"alias_kind":"pith_short_8","alias_value":"D4HJW6HH","created_at":"2026-06-11T01:10:41Z"}],"graph_snapshots":[{"event_id":"sha256:a96909cadcfad39b69b5566fd0ff897538ffd3de48a4c765d0a14248f2d9e86e","target":"graph","created_at":"2026-06-11T01:10:41Z","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":[],"endpoint":"/pith/2606.11987/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Lifted product codes are an important family of quantum low-density parity-check (QLDPC) codes, as they were the first QLDPC code family shown to be asymptotically good. Understanding the structure of their parity-check matrices $H_{\\mathsf{X}}$ and $H_{\\mathsf{Z}}$, as well as the associated Tanner graphs, is essential for analyzing their decoding behavior and error-floor performance. In this work, we show that the Tanner graphs of $H_{\\mathsf{X}}$ and $H_{\\mathsf{Z}}$ are indeed isomorphic, and investigate their graph-theoretical structure. We establish conditions ensuring the connectivity o","authors_text":"Kirsten D. Morris, Patricija \\v{S}apokait\\.e, Ragnar Freij-Hollanti","cross_cats":["math.CO","math.IT"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.IT","submitted_at":"2026-06-10T12:11:25Z","title":"Graphical Analysis of Lifted Product Code Constructions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.11987","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:fb391c56a45d36c9a405d876e0c7b016253783d9046856b8afafa15fd2db6a62","target":"record","created_at":"2026-06-11T01:10:41Z","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":"1260ef5e6fe52d373fcb73b3dc507b28a987c9ea55b74055f5f7b2646e50d590","cross_cats_sorted":["math.CO","math.IT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.IT","submitted_at":"2026-06-10T12:11:25Z","title_canon_sha256":"dfc0a79afdc220ef30be0ed1ea33bf240efbdd4d333d617d336a6b9d2d0f26a7"},"schema_version":"1.0","source":{"id":"2606.11987","kind":"arxiv","version":1}},"canonical_sha256":"1f0e9b78e7f53460e6de984c6cbaeef99884c9a79fb9a1a35aa291a888e81128","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1f0e9b78e7f53460e6de984c6cbaeef99884c9a79fb9a1a35aa291a888e81128","first_computed_at":"2026-06-11T01:10:41.189761Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-11T01:10:41.189761Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Sasr3OZxzlYHNiy10HKAryJ1iaROveC0kA0oGnZA5P8A29nvKluUPojXz4Z3N+LSpeysSVPRhnIxBORnouLVDw==","signature_status":"signed_v1","signed_at":"2026-06-11T01:10:41.190634Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.11987","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fb391c56a45d36c9a405d876e0c7b016253783d9046856b8afafa15fd2db6a62","sha256:a96909cadcfad39b69b5566fd0ff897538ffd3de48a4c765d0a14248f2d9e86e"],"state_sha256":"867ff2ba2d5435eada129a3d58921c8265d428323a263d7090fb0f454185c8e5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"II0quirnn47jqzll4x7nMVyeJKdHC26912kfCAQm8yj7Rk5ULSDWKmKZQQyDJrgnIZKI+wFklUajO+6tSZpqAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T19:23:45.401621Z","bundle_sha256":"4e11ae68ef03640d05f0236c57a16b7557db1d81854dc8dd9bfe6dc103c3ef8b"}}