{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:CTIVQNPX2FY4JS4E4Y5WIQMBHE","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":"aa54249b5e1ae5edaf6c7ff0fee493b68ac73b525876772523e8899a97c77f5c","cross_cats_sorted":["math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2014-01-14T06:03:54Z","title_canon_sha256":"87d907cbbfafb1abb79b7c1f69e5e2cb2ca226fabfda9c43a77240f0b3a5625a"},"schema_version":"1.0","source":{"id":"1401.3075","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1401.3075","created_at":"2026-05-18T01:04:04Z"},{"alias_kind":"arxiv_version","alias_value":"1401.3075v2","created_at":"2026-05-18T01:04:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1401.3075","created_at":"2026-05-18T01:04:04Z"},{"alias_kind":"pith_short_12","alias_value":"CTIVQNPX2FY4","created_at":"2026-05-18T12:28:22Z"},{"alias_kind":"pith_short_16","alias_value":"CTIVQNPX2FY4JS4E","created_at":"2026-05-18T12:28:22Z"},{"alias_kind":"pith_short_8","alias_value":"CTIVQNPX","created_at":"2026-05-18T12:28:22Z"}],"graph_snapshots":[{"event_id":"sha256:b520b3b83bcc58e7202aae92684db3ea3e60150617b2060b4318092cb7a474e3","target":"graph","created_at":"2026-05-18T01:04:04Z","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":"In an acyclic multicast network, it is well known that a linear network coding solution over GF($q$) exists when $q$ is sufficiently large. In particular, for each prime power $q$ no smaller than the number of receivers, a linear solution over GF($q$) can be efficiently constructed. In this work, we reveal that a linear solution over a given finite field does \\emph{not} necessarily imply the existence of a linear solution over all larger finite fields. Specifically, we prove by construction that: (i) For every source dimension no smaller than 3, there is a multicast network linearly solvable o","authors_text":"Keping Long, Qifu (Tyler) Sun, Xunrui Yin, Zongpeng Li","cross_cats":["math.IT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2014-01-14T06:03:54Z","title":"Multicast Network Coding and Field Sizes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.3075","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":"sha256:20aaebffbfb34301c9a0b2988591dbd5c8097ce9a1d80d6cceac0df987467fb0","target":"record","created_at":"2026-05-18T01:04:04Z","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":"aa54249b5e1ae5edaf6c7ff0fee493b68ac73b525876772523e8899a97c77f5c","cross_cats_sorted":["math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2014-01-14T06:03:54Z","title_canon_sha256":"87d907cbbfafb1abb79b7c1f69e5e2cb2ca226fabfda9c43a77240f0b3a5625a"},"schema_version":"1.0","source":{"id":"1401.3075","kind":"arxiv","version":2}},"canonical_sha256":"14d15835f7d171c4cb84e63b6441813912019a69e0b03e8d32035a99a97c3b5f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"14d15835f7d171c4cb84e63b6441813912019a69e0b03e8d32035a99a97c3b5f","first_computed_at":"2026-05-18T01:04:04.370090Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:04:04.370090Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ANFp6gPnp9Q+m1mEZWWOcCw1BlMrHu7AWgQsvq3TvFnE+wNMLIxG2O+de5dEy+rXxgyslJUjOtzXFa0x4zePCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:04:04.370569Z","signed_message":"canonical_sha256_bytes"},"source_id":"1401.3075","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:20aaebffbfb34301c9a0b2988591dbd5c8097ce9a1d80d6cceac0df987467fb0","sha256:b520b3b83bcc58e7202aae92684db3ea3e60150617b2060b4318092cb7a474e3"],"state_sha256":"455d19bd8d21cec6e8fe0e349e98fb2e621292f6781cd7ebdada478a1875f62a"}