{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:N6M7GJ4YNCCOGB3TOXC26VCOBV","short_pith_number":"pith:N6M7GJ4Y","canonical_record":{"source":{"id":"1709.05970","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2017-09-15T08:40:59Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"716846683d866bad48731687a4cd4d6312025b37d4e51aea3bbc433f336cc9c8","abstract_canon_sha256":"8970548729287f0d9197ebbdf44845e460038874f8c86a38ceb59dd7bf4e53d9"},"schema_version":"1.0"},"canonical_sha256":"6f99f327986884e3077375c5af544e0d43942f881d940d247163bfd949c973de","source":{"kind":"arxiv","id":"1709.05970","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1709.05970","created_at":"2026-05-18T00:34:58Z"},{"alias_kind":"arxiv_version","alias_value":"1709.05970v1","created_at":"2026-05-18T00:34:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.05970","created_at":"2026-05-18T00:34:58Z"},{"alias_kind":"pith_short_12","alias_value":"N6M7GJ4YNCCO","created_at":"2026-05-18T12:31:31Z"},{"alias_kind":"pith_short_16","alias_value":"N6M7GJ4YNCCOGB3T","created_at":"2026-05-18T12:31:31Z"},{"alias_kind":"pith_short_8","alias_value":"N6M7GJ4Y","created_at":"2026-05-18T12:31:31Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:N6M7GJ4YNCCOGB3TOXC26VCOBV","target":"record","payload":{"canonical_record":{"source":{"id":"1709.05970","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2017-09-15T08:40:59Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"716846683d866bad48731687a4cd4d6312025b37d4e51aea3bbc433f336cc9c8","abstract_canon_sha256":"8970548729287f0d9197ebbdf44845e460038874f8c86a38ceb59dd7bf4e53d9"},"schema_version":"1.0"},"canonical_sha256":"6f99f327986884e3077375c5af544e0d43942f881d940d247163bfd949c973de","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:34:58.614708Z","signature_b64":"mOpYgApETmILrMxp2tAIzUs1G4Ezo1vtPWr31oKGpJqVlpr6d/WQuZ+WeDeTGnUUu1KRnglWqQFEE4e9uW1vBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6f99f327986884e3077375c5af544e0d43942f881d940d247163bfd949c973de","last_reissued_at":"2026-05-18T00:34:58.613942Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:34:58.613942Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1709.05970","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":"2026-05-18T00:34:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"P9LMuNud1DvNgJO7DiusHdwSZswJP6UnYRP677OrOxcMYHMUxcHfUgqYVI8vIDOTIXu8+HG8hHYQnjs1WCBgDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T22:10:03.757646Z"},"content_sha256":"688313e2034e8b67c686c22c2fafca9d556b7637908d4e6f0230962a6eda9d4a","schema_version":"1.0","event_id":"sha256:688313e2034e8b67c686c22c2fafca9d556b7637908d4e6f0230962a6eda9d4a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:N6M7GJ4YNCCOGB3TOXC26VCOBV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the Dependence of Linear Coding Rates on the Characteristic of the Finite Field","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Brijesh Kumar Rai, Niladri Das","submitted_at":"2017-09-15T08:40:59Z","abstract_excerpt":"It is known that for any finite/co-finite set of primes there exists a network which has a rate $1$ solution if and only if the characteristic of the finite field belongs to the given set. We generalize this result to show that for any positive rational number $k/n$, and for any given finite/co-finite set of primes, there exists a network which has a rate $k/n$ fractional linear network coding solution if and only if the characteristic of the finite field belongs to the given set. For this purpose we construct two networks: $\\mathcal{N}_1$ and $\\mathcal{N}_2$; the network $\\mathcal{N}_1$ has a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.05970","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":"2026-05-18T00:34:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aA3AZpOkAgScimM/ISnnW+kOhuGujLkeTJlTXmIeQhNh9DkS/wvY9VZ8Ri8j5jYCMTzfV0BZoQB/BfajHCXGDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T22:10:03.758003Z"},"content_sha256":"ba3947433d9cf26dd3d1a29b36789144506a977ac520731394c7a1f135205403","schema_version":"1.0","event_id":"sha256:ba3947433d9cf26dd3d1a29b36789144506a977ac520731394c7a1f135205403"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/N6M7GJ4YNCCOGB3TOXC26VCOBV/bundle.json","state_url":"https://pith.science/pith/N6M7GJ4YNCCOGB3TOXC26VCOBV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/N6M7GJ4YNCCOGB3TOXC26VCOBV/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":"2026-06-30T22:10:03Z","links":{"resolver":"https://pith.science/pith/N6M7GJ4YNCCOGB3TOXC26VCOBV","bundle":"https://pith.science/pith/N6M7GJ4YNCCOGB3TOXC26VCOBV/bundle.json","state":"https://pith.science/pith/N6M7GJ4YNCCOGB3TOXC26VCOBV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/N6M7GJ4YNCCOGB3TOXC26VCOBV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:N6M7GJ4YNCCOGB3TOXC26VCOBV","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":"8970548729287f0d9197ebbdf44845e460038874f8c86a38ceb59dd7bf4e53d9","cross_cats_sorted":["math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2017-09-15T08:40:59Z","title_canon_sha256":"716846683d866bad48731687a4cd4d6312025b37d4e51aea3bbc433f336cc9c8"},"schema_version":"1.0","source":{"id":"1709.05970","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1709.05970","created_at":"2026-05-18T00:34:58Z"},{"alias_kind":"arxiv_version","alias_value":"1709.05970v1","created_at":"2026-05-18T00:34:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.05970","created_at":"2026-05-18T00:34:58Z"},{"alias_kind":"pith_short_12","alias_value":"N6M7GJ4YNCCO","created_at":"2026-05-18T12:31:31Z"},{"alias_kind":"pith_short_16","alias_value":"N6M7GJ4YNCCOGB3T","created_at":"2026-05-18T12:31:31Z"},{"alias_kind":"pith_short_8","alias_value":"N6M7GJ4Y","created_at":"2026-05-18T12:31:31Z"}],"graph_snapshots":[{"event_id":"sha256:ba3947433d9cf26dd3d1a29b36789144506a977ac520731394c7a1f135205403","target":"graph","created_at":"2026-05-18T00:34:58Z","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":"It is known that for any finite/co-finite set of primes there exists a network which has a rate $1$ solution if and only if the characteristic of the finite field belongs to the given set. We generalize this result to show that for any positive rational number $k/n$, and for any given finite/co-finite set of primes, there exists a network which has a rate $k/n$ fractional linear network coding solution if and only if the characteristic of the finite field belongs to the given set. For this purpose we construct two networks: $\\mathcal{N}_1$ and $\\mathcal{N}_2$; the network $\\mathcal{N}_1$ has a","authors_text":"Brijesh Kumar Rai, Niladri Das","cross_cats":["math.IT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2017-09-15T08:40:59Z","title":"On the Dependence of Linear Coding Rates on the Characteristic of the Finite Field"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.05970","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":"sha256:688313e2034e8b67c686c22c2fafca9d556b7637908d4e6f0230962a6eda9d4a","target":"record","created_at":"2026-05-18T00:34:58Z","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":"8970548729287f0d9197ebbdf44845e460038874f8c86a38ceb59dd7bf4e53d9","cross_cats_sorted":["math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2017-09-15T08:40:59Z","title_canon_sha256":"716846683d866bad48731687a4cd4d6312025b37d4e51aea3bbc433f336cc9c8"},"schema_version":"1.0","source":{"id":"1709.05970","kind":"arxiv","version":1}},"canonical_sha256":"6f99f327986884e3077375c5af544e0d43942f881d940d247163bfd949c973de","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6f99f327986884e3077375c5af544e0d43942f881d940d247163bfd949c973de","first_computed_at":"2026-05-18T00:34:58.613942Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:34:58.613942Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"mOpYgApETmILrMxp2tAIzUs1G4Ezo1vtPWr31oKGpJqVlpr6d/WQuZ+WeDeTGnUUu1KRnglWqQFEE4e9uW1vBw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:34:58.614708Z","signed_message":"canonical_sha256_bytes"},"source_id":"1709.05970","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:688313e2034e8b67c686c22c2fafca9d556b7637908d4e6f0230962a6eda9d4a","sha256:ba3947433d9cf26dd3d1a29b36789144506a977ac520731394c7a1f135205403"],"state_sha256":"096c0b259351f7661ce6b269617d33d4e1819cc346eb65d1a3c46d03c3c3d35c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"D+yFfV7x/CiZNxlkkIxGhbY9muHbbQ3KiFTcY/fAdlSV9vttq3Y08IJNcuOPgR55B14vr2Vb9cdoMvwdmJKFAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-30T22:10:03.759965Z","bundle_sha256":"d93fe15a08a4a55020ec9eae17606a975efca102a4ce47d1217a26d8434a1bc3"}}