{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2000:7UVNPDPMS2GPT4OO6SMS3S3DGL","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":"c80ce4fde9d0dd1371c76efcd262baf9c566c0f4c9f0bc3d614b335a7e9bb1cd","cross_cats_sorted":["cs.IT","math-ph","math.IT","math.MP"],"license":"","primary_cat":"quant-ph","submitted_at":"2000-12-20T16:20:53Z","title_canon_sha256":"1ff163f0b0b596ab2513053f0e869b6e66bb2f540b756ebc5c7cf5d5fc534d49"},"schema_version":"1.0","source":{"id":"quant-ph/0012111","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"quant-ph/0012111","created_at":"2026-05-18T03:24:20Z"},{"alias_kind":"arxiv_version","alias_value":"quant-ph/0012111v1","created_at":"2026-05-18T03:24:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.quant-ph/0012111","created_at":"2026-05-18T03:24:20Z"},{"alias_kind":"pith_short_12","alias_value":"7UVNPDPMS2GP","created_at":"2026-05-18T12:25:49Z"},{"alias_kind":"pith_short_16","alias_value":"7UVNPDPMS2GPT4OO","created_at":"2026-05-18T12:25:49Z"},{"alias_kind":"pith_short_8","alias_value":"7UVNPDPM","created_at":"2026-05-18T12:25:49Z"}],"graph_snapshots":[{"event_id":"sha256:38c4f2c2bca13de95ed5ce568f4afbad17671946596859fede5b962f4b892773","target":"graph","created_at":"2026-05-18T03:24:20Z","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"},"paper":{"abstract_excerpt":"We present a construction scheme for quantum error correcting codes. The basic ingredients are a graph and a finite abelian group, from which the code can explicitly be obtained. We prove necessary and sufficient conditions for the graph such that the resulting code corrects a certain number of errors. This allows a simple verification of the 1-error correcting property of fivefold codes in any dimension. As new examples we construct a large class of codes saturating the singleton bound, as well as a tenfold code detecting 3 errors.","authors_text":"D. Schlingemann, R.F. Werner","cross_cats":["cs.IT","math-ph","math.IT","math.MP"],"headline":"","license":"","primary_cat":"quant-ph","submitted_at":"2000-12-20T16:20:53Z","title":"Quantum error-correcting codes associated with graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"quant-ph/0012111","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":"sha256:377dcf2909e2e9c5ec58731136a7d00f8ae462888ef0c6b0cd12ee9660a3d9f6","target":"record","created_at":"2026-05-18T03:24:20Z","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":"c80ce4fde9d0dd1371c76efcd262baf9c566c0f4c9f0bc3d614b335a7e9bb1cd","cross_cats_sorted":["cs.IT","math-ph","math.IT","math.MP"],"license":"","primary_cat":"quant-ph","submitted_at":"2000-12-20T16:20:53Z","title_canon_sha256":"1ff163f0b0b596ab2513053f0e869b6e66bb2f540b756ebc5c7cf5d5fc534d49"},"schema_version":"1.0","source":{"id":"quant-ph/0012111","kind":"arxiv","version":1}},"canonical_sha256":"fd2ad78dec968cf9f1cef4992dcb6332e2348d0dc2c187fc4e9b7f4daa744300","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fd2ad78dec968cf9f1cef4992dcb6332e2348d0dc2c187fc4e9b7f4daa744300","first_computed_at":"2026-05-18T03:24:20.611346Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:24:20.611346Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LodSRGwNLSTQ4jid3n5FEcpttxu1tmmXZVMgvPprvHAGrREhxNtpsVDoCkNDFHTLK3okjFH89kqJp6hEIswdCA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:24:20.611889Z","signed_message":"canonical_sha256_bytes"},"source_id":"quant-ph/0012111","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:377dcf2909e2e9c5ec58731136a7d00f8ae462888ef0c6b0cd12ee9660a3d9f6","sha256:38c4f2c2bca13de95ed5ce568f4afbad17671946596859fede5b962f4b892773"],"state_sha256":"38da0d5ba45e0c06132eb1edad5aa9cd947e4646a935f43dffe3f141250aa950"}