{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:7Q3CL57ZIZ5BUNZ4FYK3QC5RKV","short_pith_number":"pith:7Q3CL57Z","canonical_record":{"source":{"id":"1212.5679","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2012-12-22T10:30:50Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"e8bcf5988993f7327e50cfd8bfe0bb8f875157fcad7aa44bce5d73284200fe93","abstract_canon_sha256":"9bb57cd40b8b58573e285f0c5cc90ec1740e58337f20109d4b17e76930e61b2b"},"schema_version":"1.0"},"canonical_sha256":"fc3625f7f9467a1a373c2e15b80bb155710ac306073557aceb39a84d29d4179c","source":{"kind":"arxiv","id":"1212.5679","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.5679","created_at":"2026-05-18T03:37:51Z"},{"alias_kind":"arxiv_version","alias_value":"1212.5679v1","created_at":"2026-05-18T03:37:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.5679","created_at":"2026-05-18T03:37:51Z"},{"alias_kind":"pith_short_12","alias_value":"7Q3CL57ZIZ5B","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_16","alias_value":"7Q3CL57ZIZ5BUNZ4","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_8","alias_value":"7Q3CL57Z","created_at":"2026-05-18T12:26:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:7Q3CL57ZIZ5BUNZ4FYK3QC5RKV","target":"record","payload":{"canonical_record":{"source":{"id":"1212.5679","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2012-12-22T10:30:50Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"e8bcf5988993f7327e50cfd8bfe0bb8f875157fcad7aa44bce5d73284200fe93","abstract_canon_sha256":"9bb57cd40b8b58573e285f0c5cc90ec1740e58337f20109d4b17e76930e61b2b"},"schema_version":"1.0"},"canonical_sha256":"fc3625f7f9467a1a373c2e15b80bb155710ac306073557aceb39a84d29d4179c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:37:51.806922Z","signature_b64":"dMUtIOyHoq6gGhcl55DgQYTx5Hi4Nbnxn6f9Gxs4SaSSbAQUOfsKUjvJ0eEXKADDEP6r5bQVO6J0+Ze8O0rwCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fc3625f7f9467a1a373c2e15b80bb155710ac306073557aceb39a84d29d4179c","last_reissued_at":"2026-05-18T03:37:51.806059Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:37:51.806059Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1212.5679","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":"2026-05-18T03:37:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"b5xJbr3n2HTaEvRHpCzY4W76zb4GI5nFE+WIxAN44lS/nfP/CE44F8zk1DYBE++y8hp+KJvvyTxFxTj+RMVcBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T19:35:16.856908Z"},"content_sha256":"d6483f96c1d14ce64f067deacd5bcd842eaf431f07d4f6788c3536fe43bd54a7","schema_version":"1.0","event_id":"sha256:d6483f96c1d14ce64f067deacd5bcd842eaf431f07d4f6788c3536fe43bd54a7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:7Q3CL57ZIZ5BUNZ4FYK3QC5RKV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Cumulative Distance Enumerators of Random Codes and their Thresholds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Chaoping Xing, Hongwei Liu, Jing Shen, San Ling, Yun Fan","submitted_at":"2012-12-22T10:30:50Z","abstract_excerpt":"Cumulative weight enumerators of random linear codes are introduced, their asymptotic properties are studied, and very sharp thresholds are exhibited; as a consequence, it is shown that the asymptotic Gilbert-Varshamov bound is a very sharp threshold point for the density of the linear codes whose relative distance is greater than a given positive number. For arbitrary random codes, similar settings and results are exhibited; in particular, the very sharp threshold point for the density of the codes whose relative distance is greater than a given positive number is located at half the asymptot"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.5679","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":"2026-05-18T03:37:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Eawe6nJWqqpoI8fSVI373Kh2mIg+pbdoOMJVxKAnuficOv1q79TyeKdbRFQyOk1cgiKpSpA5zYxNIVacKHMsBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T19:35:16.857589Z"},"content_sha256":"d24e2b80f6d4ef2db3e722e2f1f8fa55f30cbc40b74ff827fa95c967eb21bbb3","schema_version":"1.0","event_id":"sha256:d24e2b80f6d4ef2db3e722e2f1f8fa55f30cbc40b74ff827fa95c967eb21bbb3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7Q3CL57ZIZ5BUNZ4FYK3QC5RKV/bundle.json","state_url":"https://pith.science/pith/7Q3CL57ZIZ5BUNZ4FYK3QC5RKV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7Q3CL57ZIZ5BUNZ4FYK3QC5RKV/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":"2026-06-08T19:35:16Z","links":{"resolver":"https://pith.science/pith/7Q3CL57ZIZ5BUNZ4FYK3QC5RKV","bundle":"https://pith.science/pith/7Q3CL57ZIZ5BUNZ4FYK3QC5RKV/bundle.json","state":"https://pith.science/pith/7Q3CL57ZIZ5BUNZ4FYK3QC5RKV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7Q3CL57ZIZ5BUNZ4FYK3QC5RKV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:7Q3CL57ZIZ5BUNZ4FYK3QC5RKV","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":"9bb57cd40b8b58573e285f0c5cc90ec1740e58337f20109d4b17e76930e61b2b","cross_cats_sorted":["math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2012-12-22T10:30:50Z","title_canon_sha256":"e8bcf5988993f7327e50cfd8bfe0bb8f875157fcad7aa44bce5d73284200fe93"},"schema_version":"1.0","source":{"id":"1212.5679","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.5679","created_at":"2026-05-18T03:37:51Z"},{"alias_kind":"arxiv_version","alias_value":"1212.5679v1","created_at":"2026-05-18T03:37:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.5679","created_at":"2026-05-18T03:37:51Z"},{"alias_kind":"pith_short_12","alias_value":"7Q3CL57ZIZ5B","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_16","alias_value":"7Q3CL57ZIZ5BUNZ4","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_8","alias_value":"7Q3CL57Z","created_at":"2026-05-18T12:26:58Z"}],"graph_snapshots":[{"event_id":"sha256:d24e2b80f6d4ef2db3e722e2f1f8fa55f30cbc40b74ff827fa95c967eb21bbb3","target":"graph","created_at":"2026-05-18T03:37:51Z","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":"Cumulative weight enumerators of random linear codes are introduced, their asymptotic properties are studied, and very sharp thresholds are exhibited; as a consequence, it is shown that the asymptotic Gilbert-Varshamov bound is a very sharp threshold point for the density of the linear codes whose relative distance is greater than a given positive number. For arbitrary random codes, similar settings and results are exhibited; in particular, the very sharp threshold point for the density of the codes whose relative distance is greater than a given positive number is located at half the asymptot","authors_text":"Chaoping Xing, Hongwei Liu, Jing Shen, San Ling, Yun Fan","cross_cats":["math.IT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2012-12-22T10:30:50Z","title":"Cumulative Distance Enumerators of Random Codes and their Thresholds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.5679","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:d6483f96c1d14ce64f067deacd5bcd842eaf431f07d4f6788c3536fe43bd54a7","target":"record","created_at":"2026-05-18T03:37:51Z","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":"9bb57cd40b8b58573e285f0c5cc90ec1740e58337f20109d4b17e76930e61b2b","cross_cats_sorted":["math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2012-12-22T10:30:50Z","title_canon_sha256":"e8bcf5988993f7327e50cfd8bfe0bb8f875157fcad7aa44bce5d73284200fe93"},"schema_version":"1.0","source":{"id":"1212.5679","kind":"arxiv","version":1}},"canonical_sha256":"fc3625f7f9467a1a373c2e15b80bb155710ac306073557aceb39a84d29d4179c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fc3625f7f9467a1a373c2e15b80bb155710ac306073557aceb39a84d29d4179c","first_computed_at":"2026-05-18T03:37:51.806059Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:37:51.806059Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"dMUtIOyHoq6gGhcl55DgQYTx5Hi4Nbnxn6f9Gxs4SaSSbAQUOfsKUjvJ0eEXKADDEP6r5bQVO6J0+Ze8O0rwCA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:37:51.806922Z","signed_message":"canonical_sha256_bytes"},"source_id":"1212.5679","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d6483f96c1d14ce64f067deacd5bcd842eaf431f07d4f6788c3536fe43bd54a7","sha256:d24e2b80f6d4ef2db3e722e2f1f8fa55f30cbc40b74ff827fa95c967eb21bbb3"],"state_sha256":"1fb223cb8ee9ae432bacbc9e180b5e01e3c8a18b0409a06bb7f107df5bea2c2e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bl3WTGBkpEpQje7U3pwoV5ru85f63qzbQripknAVfLMvxcc5a0E7p2IilZWD3K3ldqOsx/ivK1gTxv3Ro/1zBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T19:35:16.861639Z","bundle_sha256":"e1a3110278af834f5a4d2982be244895a60bc31113d5c7350857ba41a6561ec2"}}