{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:XYN25NSULLY2NSGKEXU3XWPHOB","short_pith_number":"pith:XYN25NSU","canonical_record":{"source":{"id":"1501.06543","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2015-01-26T19:56:41Z","cross_cats_sorted":["cs.CR","cs.DM","math.CO","math.IT","math.NT"],"title_canon_sha256":"c49b224701d6d0247e06337924f573ebdeb31062f8cfec839e1382539201ecd8","abstract_canon_sha256":"a3e0ae5c6a92744b9980417845c6459e38b68e3b405e2298edde5554a61c27bc"},"schema_version":"1.0"},"canonical_sha256":"be1baeb6545af1a6c8ca25e9bbd9e7707a38d5a3040875ece18f2e79da7f3de3","source":{"kind":"arxiv","id":"1501.06543","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.06543","created_at":"2026-05-18T02:28:41Z"},{"alias_kind":"arxiv_version","alias_value":"1501.06543v1","created_at":"2026-05-18T02:28:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.06543","created_at":"2026-05-18T02:28:41Z"},{"alias_kind":"pith_short_12","alias_value":"XYN25NSULLY2","created_at":"2026-05-18T12:29:50Z"},{"alias_kind":"pith_short_16","alias_value":"XYN25NSULLY2NSGK","created_at":"2026-05-18T12:29:50Z"},{"alias_kind":"pith_short_8","alias_value":"XYN25NSU","created_at":"2026-05-18T12:29:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:XYN25NSULLY2NSGKEXU3XWPHOB","target":"record","payload":{"canonical_record":{"source":{"id":"1501.06543","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2015-01-26T19:56:41Z","cross_cats_sorted":["cs.CR","cs.DM","math.CO","math.IT","math.NT"],"title_canon_sha256":"c49b224701d6d0247e06337924f573ebdeb31062f8cfec839e1382539201ecd8","abstract_canon_sha256":"a3e0ae5c6a92744b9980417845c6459e38b68e3b405e2298edde5554a61c27bc"},"schema_version":"1.0"},"canonical_sha256":"be1baeb6545af1a6c8ca25e9bbd9e7707a38d5a3040875ece18f2e79da7f3de3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:28:41.110944Z","signature_b64":"7y3dpItgKdzt+kb2BaxHkYdqvPZBdMWN9JzvAMkFR//Lp/98ZhsE3IRlGl5+O1FVLUV+XE4CMvM9dfMKV7NYBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"be1baeb6545af1a6c8ca25e9bbd9e7707a38d5a3040875ece18f2e79da7f3de3","last_reissued_at":"2026-05-18T02:28:41.110485Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:28:41.110485Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1501.06543","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":"2026-05-18T02:28:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kjT+dfmeyTMlxliZ7zGCIXof1joLwga1pG2+WRVws0WusxIbg/BDrlelEKDJ/V7Q4MTabfAEu1GuGzN7vATyBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T02:24:20.092567Z"},"content_sha256":"bcf7e213e871d504c4d6b35a40645f6dcd1306ee106b699b0b3c843583d4bc9c","schema_version":"1.0","event_id":"sha256:bcf7e213e871d504c4d6b35a40645f6dcd1306ee106b699b0b3c843583d4bc9c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:XYN25NSULLY2NSGKEXU3XWPHOB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Construction of Quasi-Cyclic Product Codes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CR","cs.DM","math.CO","math.IT","math.NT"],"primary_cat":"cs.IT","authors_text":"Alexander Zeh, San Ling","submitted_at":"2015-01-26T19:56:41Z","abstract_excerpt":"Linear quasi-cyclic product codes over finite fields are investigated. Given the generating set in the form of a reduced Gr{\\\"o}bner basis of a quasi-cyclic component code and the generator polynomial of a second cyclic component code, an explicit expression of the basis of the generating set of the quasi-cyclic product code is given. Furthermore, the reduced Gr{\\\"o}bner basis of a one-level quasi-cyclic product code is derived."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.06543","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":"2026-05-18T02:28:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"R9aidNkkNWvSC5bVjVLwtQqu5M3R7phJaLcXZ+vYgUfWYbHwm5BvF9tA8yPsuWVdlKLVLeplEDfRu8jHDjOGAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T02:24:20.092915Z"},"content_sha256":"f10fadc1c14b55cbb50d64c302c66fa02103388ece50f09c013dedaf0fa7a80f","schema_version":"1.0","event_id":"sha256:f10fadc1c14b55cbb50d64c302c66fa02103388ece50f09c013dedaf0fa7a80f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XYN25NSULLY2NSGKEXU3XWPHOB/bundle.json","state_url":"https://pith.science/pith/XYN25NSULLY2NSGKEXU3XWPHOB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XYN25NSULLY2NSGKEXU3XWPHOB/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":"2026-06-21T02:24:20Z","links":{"resolver":"https://pith.science/pith/XYN25NSULLY2NSGKEXU3XWPHOB","bundle":"https://pith.science/pith/XYN25NSULLY2NSGKEXU3XWPHOB/bundle.json","state":"https://pith.science/pith/XYN25NSULLY2NSGKEXU3XWPHOB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XYN25NSULLY2NSGKEXU3XWPHOB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:XYN25NSULLY2NSGKEXU3XWPHOB","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":"a3e0ae5c6a92744b9980417845c6459e38b68e3b405e2298edde5554a61c27bc","cross_cats_sorted":["cs.CR","cs.DM","math.CO","math.IT","math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2015-01-26T19:56:41Z","title_canon_sha256":"c49b224701d6d0247e06337924f573ebdeb31062f8cfec839e1382539201ecd8"},"schema_version":"1.0","source":{"id":"1501.06543","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.06543","created_at":"2026-05-18T02:28:41Z"},{"alias_kind":"arxiv_version","alias_value":"1501.06543v1","created_at":"2026-05-18T02:28:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.06543","created_at":"2026-05-18T02:28:41Z"},{"alias_kind":"pith_short_12","alias_value":"XYN25NSULLY2","created_at":"2026-05-18T12:29:50Z"},{"alias_kind":"pith_short_16","alias_value":"XYN25NSULLY2NSGK","created_at":"2026-05-18T12:29:50Z"},{"alias_kind":"pith_short_8","alias_value":"XYN25NSU","created_at":"2026-05-18T12:29:50Z"}],"graph_snapshots":[{"event_id":"sha256:f10fadc1c14b55cbb50d64c302c66fa02103388ece50f09c013dedaf0fa7a80f","target":"graph","created_at":"2026-05-18T02:28:41Z","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"},"paper":{"abstract_excerpt":"Linear quasi-cyclic product codes over finite fields are investigated. Given the generating set in the form of a reduced Gr{\\\"o}bner basis of a quasi-cyclic component code and the generator polynomial of a second cyclic component code, an explicit expression of the basis of the generating set of the quasi-cyclic product code is given. Furthermore, the reduced Gr{\\\"o}bner basis of a one-level quasi-cyclic product code is derived.","authors_text":"Alexander Zeh, San Ling","cross_cats":["cs.CR","cs.DM","math.CO","math.IT","math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2015-01-26T19:56:41Z","title":"Construction of Quasi-Cyclic Product Codes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.06543","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:bcf7e213e871d504c4d6b35a40645f6dcd1306ee106b699b0b3c843583d4bc9c","target":"record","created_at":"2026-05-18T02:28:41Z","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":"a3e0ae5c6a92744b9980417845c6459e38b68e3b405e2298edde5554a61c27bc","cross_cats_sorted":["cs.CR","cs.DM","math.CO","math.IT","math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2015-01-26T19:56:41Z","title_canon_sha256":"c49b224701d6d0247e06337924f573ebdeb31062f8cfec839e1382539201ecd8"},"schema_version":"1.0","source":{"id":"1501.06543","kind":"arxiv","version":1}},"canonical_sha256":"be1baeb6545af1a6c8ca25e9bbd9e7707a38d5a3040875ece18f2e79da7f3de3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"be1baeb6545af1a6c8ca25e9bbd9e7707a38d5a3040875ece18f2e79da7f3de3","first_computed_at":"2026-05-18T02:28:41.110485Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:28:41.110485Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7y3dpItgKdzt+kb2BaxHkYdqvPZBdMWN9JzvAMkFR//Lp/98ZhsE3IRlGl5+O1FVLUV+XE4CMvM9dfMKV7NYBw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:28:41.110944Z","signed_message":"canonical_sha256_bytes"},"source_id":"1501.06543","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bcf7e213e871d504c4d6b35a40645f6dcd1306ee106b699b0b3c843583d4bc9c","sha256:f10fadc1c14b55cbb50d64c302c66fa02103388ece50f09c013dedaf0fa7a80f"],"state_sha256":"39dafc6d8d8fbbaa708fd23e917535c678b5fe104d812665914bb653fd2c4e2c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qNEDCr8qcMhqmAXUmFLh27v/xYTtFNcGrgDm3Sm/hO0AoOxIYp6ESs48UeQUGLwnCu5HC9vMNYcW4CkbS2G0Cw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-21T02:24:20.094975Z","bundle_sha256":"55e5fb82b9db597583db2f113a834d3730801b5f644e43db892d62a275b8b342"}}