{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:WXIUYWP43VR66V7YSCYILGXA5B","short_pith_number":"pith:WXIUYWP4","canonical_record":{"source":{"id":"1608.01737","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-08-05T01:52:17Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"eb0c039d1d1aac4c5ee0c4296840732e5a92a3c5042ad23449db3c04b5841266","abstract_canon_sha256":"a0a705e0ad22d0bf542ad3fe69851bf03086c0c668d54157f1c776d1499ad795"},"schema_version":"1.0"},"canonical_sha256":"b5d14c59fcdd63ef57f890b0859ae0e857ea527c1ffb6c3a07fb6f9040222d3a","source":{"kind":"arxiv","id":"1608.01737","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.01737","created_at":"2026-05-18T00:24:54Z"},{"alias_kind":"arxiv_version","alias_value":"1608.01737v2","created_at":"2026-05-18T00:24:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.01737","created_at":"2026-05-18T00:24:54Z"},{"alias_kind":"pith_short_12","alias_value":"WXIUYWP43VR6","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"WXIUYWP43VR66V7Y","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"WXIUYWP4","created_at":"2026-05-18T12:30:51Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:WXIUYWP43VR66V7YSCYILGXA5B","target":"record","payload":{"canonical_record":{"source":{"id":"1608.01737","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-08-05T01:52:17Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"eb0c039d1d1aac4c5ee0c4296840732e5a92a3c5042ad23449db3c04b5841266","abstract_canon_sha256":"a0a705e0ad22d0bf542ad3fe69851bf03086c0c668d54157f1c776d1499ad795"},"schema_version":"1.0"},"canonical_sha256":"b5d14c59fcdd63ef57f890b0859ae0e857ea527c1ffb6c3a07fb6f9040222d3a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:24:54.509405Z","signature_b64":"QJdz7v6CytanO/lpl6kVHfs64pXoC0Dn+BUgYldSD6sWCxl3tP1cEeAfO/0Y5hx8th0NjmIg9T9Kh6zdQ3WAAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b5d14c59fcdd63ef57f890b0859ae0e857ea527c1ffb6c3a07fb6f9040222d3a","last_reissued_at":"2026-05-18T00:24:54.508714Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:24:54.508714Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1608.01737","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":"2026-05-18T00:24:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AJerARf96jgnS4nDVgViHEhfrs0qk2IGfTNghumathYLG0vFgm7sNJ8gqht+RBn5XKKRWJwYLgaCXmhDp0iMDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T14:46:35.569821Z"},"content_sha256":"c6b937a35bf5063f1d4327e207af422d037795892ed661e845f0211019ac7298","schema_version":"1.0","event_id":"sha256:c6b937a35bf5063f1d4327e207af422d037795892ed661e845f0211019ac7298"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:WXIUYWP43VR66V7YSCYILGXA5B","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Linear Network Coding over Rings, Part II: Vector Codes and Non-Commutative Alphabets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Joseph Connelly, Kenneth Zeger","submitted_at":"2016-08-05T01:52:17Z","abstract_excerpt":"We prove the following results regarding the linear solvability of networks over various alphabets. For any network, the following are equivalent: (i) vector linear solvability over some finite field, (ii) scalar linear solvability over some ring, (iii) linear solvability over some module. Analogously, the following are equivalent: (a) scalar linear solvability over some finite field, (b) scalar linear solvability over some commutative ring, (c) linear solvability over some module whose ring is commutative. Whenever any network is linearly solvable over a module, a smallest such module arises "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.01737","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":"2026-05-18T00:24:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ooh5Vku7+wi5yNXDfLQBob9CaoL4bTZUd8n4vchYdt6OM9bSIKNsC6g+uhjz/f3IKVdnFg0XYwt0aUlH3oEnAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T14:46:35.570161Z"},"content_sha256":"79d4d8c71f83988e30135abe13038013028f2d2ae3601c1be19d4ccb5769598b","schema_version":"1.0","event_id":"sha256:79d4d8c71f83988e30135abe13038013028f2d2ae3601c1be19d4ccb5769598b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WXIUYWP43VR66V7YSCYILGXA5B/bundle.json","state_url":"https://pith.science/pith/WXIUYWP43VR66V7YSCYILGXA5B/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WXIUYWP43VR66V7YSCYILGXA5B/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":"2026-06-03T14:46:35Z","links":{"resolver":"https://pith.science/pith/WXIUYWP43VR66V7YSCYILGXA5B","bundle":"https://pith.science/pith/WXIUYWP43VR66V7YSCYILGXA5B/bundle.json","state":"https://pith.science/pith/WXIUYWP43VR66V7YSCYILGXA5B/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WXIUYWP43VR66V7YSCYILGXA5B/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:WXIUYWP43VR66V7YSCYILGXA5B","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":"a0a705e0ad22d0bf542ad3fe69851bf03086c0c668d54157f1c776d1499ad795","cross_cats_sorted":["math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-08-05T01:52:17Z","title_canon_sha256":"eb0c039d1d1aac4c5ee0c4296840732e5a92a3c5042ad23449db3c04b5841266"},"schema_version":"1.0","source":{"id":"1608.01737","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.01737","created_at":"2026-05-18T00:24:54Z"},{"alias_kind":"arxiv_version","alias_value":"1608.01737v2","created_at":"2026-05-18T00:24:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.01737","created_at":"2026-05-18T00:24:54Z"},{"alias_kind":"pith_short_12","alias_value":"WXIUYWP43VR6","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"WXIUYWP43VR66V7Y","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"WXIUYWP4","created_at":"2026-05-18T12:30:51Z"}],"graph_snapshots":[{"event_id":"sha256:79d4d8c71f83988e30135abe13038013028f2d2ae3601c1be19d4ccb5769598b","target":"graph","created_at":"2026-05-18T00:24:54Z","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 prove the following results regarding the linear solvability of networks over various alphabets. For any network, the following are equivalent: (i) vector linear solvability over some finite field, (ii) scalar linear solvability over some ring, (iii) linear solvability over some module. Analogously, the following are equivalent: (a) scalar linear solvability over some finite field, (b) scalar linear solvability over some commutative ring, (c) linear solvability over some module whose ring is commutative. Whenever any network is linearly solvable over a module, a smallest such module arises ","authors_text":"Joseph Connelly, Kenneth Zeger","cross_cats":["math.IT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-08-05T01:52:17Z","title":"Linear Network Coding over Rings, Part II: Vector Codes and Non-Commutative Alphabets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.01737","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:c6b937a35bf5063f1d4327e207af422d037795892ed661e845f0211019ac7298","target":"record","created_at":"2026-05-18T00:24:54Z","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":"a0a705e0ad22d0bf542ad3fe69851bf03086c0c668d54157f1c776d1499ad795","cross_cats_sorted":["math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-08-05T01:52:17Z","title_canon_sha256":"eb0c039d1d1aac4c5ee0c4296840732e5a92a3c5042ad23449db3c04b5841266"},"schema_version":"1.0","source":{"id":"1608.01737","kind":"arxiv","version":2}},"canonical_sha256":"b5d14c59fcdd63ef57f890b0859ae0e857ea527c1ffb6c3a07fb6f9040222d3a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b5d14c59fcdd63ef57f890b0859ae0e857ea527c1ffb6c3a07fb6f9040222d3a","first_computed_at":"2026-05-18T00:24:54.508714Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:24:54.508714Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"QJdz7v6CytanO/lpl6kVHfs64pXoC0Dn+BUgYldSD6sWCxl3tP1cEeAfO/0Y5hx8th0NjmIg9T9Kh6zdQ3WAAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:24:54.509405Z","signed_message":"canonical_sha256_bytes"},"source_id":"1608.01737","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c6b937a35bf5063f1d4327e207af422d037795892ed661e845f0211019ac7298","sha256:79d4d8c71f83988e30135abe13038013028f2d2ae3601c1be19d4ccb5769598b"],"state_sha256":"9695054681b33e6394cdbde5a066af7e3473dd36dffd3b27ae097d678d4f495f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sP4PeAujA45f2xfYDxI3CYsio4hlf/PC/Ll1EymWFiYmZfBfcaq8p2Olv9gA31M+KmbSg4s4tBg+J23aMBDADQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T14:46:35.572082Z","bundle_sha256":"0517f33b8f0d84409de6d8c71f858c33440136b9e1e679e6dc5c84cadafe794b"}}