{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:JBR6K3FNIRZPPWKSXNQ2RUB26K","short_pith_number":"pith:JBR6K3FN","canonical_record":{"source":{"id":"1610.04273","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-10-13T21:58:43Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"160091117e5c302c1f542efd5ef73bfbf5f00200ee9d29f9c3525a1d1463e910","abstract_canon_sha256":"015708adc5e4dc7fb01d0914036a710a15a77ae4002f07663bfe36177073c79d"},"schema_version":"1.0"},"canonical_sha256":"4863e56cad4472f7d952bb61a8d03af2a82bf54b3f02fc6a216b43172804b59c","source":{"kind":"arxiv","id":"1610.04273","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.04273","created_at":"2026-05-18T01:01:21Z"},{"alias_kind":"arxiv_version","alias_value":"1610.04273v2","created_at":"2026-05-18T01:01:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.04273","created_at":"2026-05-18T01:01:21Z"},{"alias_kind":"pith_short_12","alias_value":"JBR6K3FNIRZP","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_16","alias_value":"JBR6K3FNIRZPPWKS","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_8","alias_value":"JBR6K3FN","created_at":"2026-05-18T12:30:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:JBR6K3FNIRZPPWKSXNQ2RUB26K","target":"record","payload":{"canonical_record":{"source":{"id":"1610.04273","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-10-13T21:58:43Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"160091117e5c302c1f542efd5ef73bfbf5f00200ee9d29f9c3525a1d1463e910","abstract_canon_sha256":"015708adc5e4dc7fb01d0914036a710a15a77ae4002f07663bfe36177073c79d"},"schema_version":"1.0"},"canonical_sha256":"4863e56cad4472f7d952bb61a8d03af2a82bf54b3f02fc6a216b43172804b59c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:01:21.835391Z","signature_b64":"m4OpIhRaclf6vM21FHq4upOYrlbFWBUdNoyClKRaPzr0xA0o9M6r/zU6rA8FJ544jK1g9VGbV4uFguSYGUJXCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4863e56cad4472f7d952bb61a8d03af2a82bf54b3f02fc6a216b43172804b59c","last_reissued_at":"2026-05-18T01:01:21.834753Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:01:21.834753Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1610.04273","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:01:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LX9BXSqWT1OEgk7fYXFpKCIwvoRiG6tj0Fs8L41onMv7JhxXFQq8ZqWMc8zr1GJQvxRJdACkjKcZ0f9qi6BxBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T16:55:02.094211Z"},"content_sha256":"55ff7ec105477ed3b28af3748eca6f2ba843b2066eb65deb7ab3bee7e85ec0e3","schema_version":"1.0","event_id":"sha256:55ff7ec105477ed3b28af3748eca6f2ba843b2066eb65deb7ab3bee7e85ec0e3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:JBR6K3FNIRZPPWKSXNQ2RUB26K","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Generalized and Extended Product Codes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Mario Blaum, Steve Hetzler","submitted_at":"2016-10-13T21:58:43Z","abstract_excerpt":"Generalized Product (GPC) Codes, an unification of Product Codes and Integrated Interleaved (II) Codes, are presented. Applications for approaches requiring local and global parities are described. The more general problem of extending product codes by adding global parities is studied and an upper bound on the minimum distance of such codes is obtained. Codes with one, two and three global parities whose minimum distances meet the bound are presented. Tradeoffs between optimality and field size are discussed."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.04273","kind":"arxiv","version":2},"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":"2026-05-18T01:01:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vwQqh2Z1JG4O6q+K+gst9sM12KTeP/Q1666W3Ho+BORtO6/r/iGInbx4+GMGV719stt27qhsY2dO0ePVglHCDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T16:55:02.094573Z"},"content_sha256":"d5017594999b8feac30ab36e13bd3a0e6cb5f6fae6e9d8342b015e18a3765b73","schema_version":"1.0","event_id":"sha256:d5017594999b8feac30ab36e13bd3a0e6cb5f6fae6e9d8342b015e18a3765b73"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JBR6K3FNIRZPPWKSXNQ2RUB26K/bundle.json","state_url":"https://pith.science/pith/JBR6K3FNIRZPPWKSXNQ2RUB26K/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JBR6K3FNIRZPPWKSXNQ2RUB26K/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":"2026-06-23T16:55:02Z","links":{"resolver":"https://pith.science/pith/JBR6K3FNIRZPPWKSXNQ2RUB26K","bundle":"https://pith.science/pith/JBR6K3FNIRZPPWKSXNQ2RUB26K/bundle.json","state":"https://pith.science/pith/JBR6K3FNIRZPPWKSXNQ2RUB26K/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JBR6K3FNIRZPPWKSXNQ2RUB26K/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:JBR6K3FNIRZPPWKSXNQ2RUB26K","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":"015708adc5e4dc7fb01d0914036a710a15a77ae4002f07663bfe36177073c79d","cross_cats_sorted":["math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-10-13T21:58:43Z","title_canon_sha256":"160091117e5c302c1f542efd5ef73bfbf5f00200ee9d29f9c3525a1d1463e910"},"schema_version":"1.0","source":{"id":"1610.04273","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.04273","created_at":"2026-05-18T01:01:21Z"},{"alias_kind":"arxiv_version","alias_value":"1610.04273v2","created_at":"2026-05-18T01:01:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.04273","created_at":"2026-05-18T01:01:21Z"},{"alias_kind":"pith_short_12","alias_value":"JBR6K3FNIRZP","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_16","alias_value":"JBR6K3FNIRZPPWKS","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_8","alias_value":"JBR6K3FN","created_at":"2026-05-18T12:30:25Z"}],"graph_snapshots":[{"event_id":"sha256:d5017594999b8feac30ab36e13bd3a0e6cb5f6fae6e9d8342b015e18a3765b73","target":"graph","created_at":"2026-05-18T01:01:21Z","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":"Generalized Product (GPC) Codes, an unification of Product Codes and Integrated Interleaved (II) Codes, are presented. Applications for approaches requiring local and global parities are described. The more general problem of extending product codes by adding global parities is studied and an upper bound on the minimum distance of such codes is obtained. Codes with one, two and three global parities whose minimum distances meet the bound are presented. Tradeoffs between optimality and field size are discussed.","authors_text":"Mario Blaum, Steve Hetzler","cross_cats":["math.IT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-10-13T21:58:43Z","title":"Generalized and Extended Product Codes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.04273","kind":"arxiv","version":2},"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":"sha256:55ff7ec105477ed3b28af3748eca6f2ba843b2066eb65deb7ab3bee7e85ec0e3","target":"record","created_at":"2026-05-18T01:01:21Z","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":"015708adc5e4dc7fb01d0914036a710a15a77ae4002f07663bfe36177073c79d","cross_cats_sorted":["math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-10-13T21:58:43Z","title_canon_sha256":"160091117e5c302c1f542efd5ef73bfbf5f00200ee9d29f9c3525a1d1463e910"},"schema_version":"1.0","source":{"id":"1610.04273","kind":"arxiv","version":2}},"canonical_sha256":"4863e56cad4472f7d952bb61a8d03af2a82bf54b3f02fc6a216b43172804b59c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4863e56cad4472f7d952bb61a8d03af2a82bf54b3f02fc6a216b43172804b59c","first_computed_at":"2026-05-18T01:01:21.834753Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:01:21.834753Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"m4OpIhRaclf6vM21FHq4upOYrlbFWBUdNoyClKRaPzr0xA0o9M6r/zU6rA8FJ544jK1g9VGbV4uFguSYGUJXCw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:01:21.835391Z","signed_message":"canonical_sha256_bytes"},"source_id":"1610.04273","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:55ff7ec105477ed3b28af3748eca6f2ba843b2066eb65deb7ab3bee7e85ec0e3","sha256:d5017594999b8feac30ab36e13bd3a0e6cb5f6fae6e9d8342b015e18a3765b73"],"state_sha256":"cbc4eb68576dace973732ad5adaa70b78b716170c7b4db523237467785e4e4dd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BdQHjPBKgEAjVRMYf7VwkiB2sczPccARHO9FfKRzN3F19EuPiOBavtMYsz/w/gcPT4IIKiANejT9uZXOEVaEBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T16:55:02.096435Z","bundle_sha256":"0c232089e7494fd8710704ac59d73859bbd0c47388a45a8c9692617ad641097d"}}