{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:CDWXMAHDRENAM66QFXMG3XRKCS","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":"d3e568e70cc36b31c32825fc73714fd494717f8151ca26d8962592c50da6bbea","cross_cats_sorted":["cs.IT","math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-06T12:13:20Z","title_canon_sha256":"70efac4092383dba57e20458a05681d6ebd50da26799054ed0f5408815d384b4"},"schema_version":"1.0","source":{"id":"1804.02219","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1804.02219","created_at":"2026-05-18T00:04:37Z"},{"alias_kind":"arxiv_version","alias_value":"1804.02219v3","created_at":"2026-05-18T00:04:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.02219","created_at":"2026-05-18T00:04:37Z"},{"alias_kind":"pith_short_12","alias_value":"CDWXMAHDRENA","created_at":"2026-05-18T12:32:16Z"},{"alias_kind":"pith_short_16","alias_value":"CDWXMAHDRENAM66Q","created_at":"2026-05-18T12:32:16Z"},{"alias_kind":"pith_short_8","alias_value":"CDWXMAHD","created_at":"2026-05-18T12:32:16Z"}],"graph_snapshots":[{"event_id":"sha256:0cd19234c051fa84978d43ca2147b4bc64e3dfb9dffe8746a202b44370698faa","target":"graph","created_at":"2026-05-18T00:04:37Z","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":"Codes in finite projective spaces equipped with the subspace distance have been proposed for error control in random linear network coding. Here we collect the present knowledge on lower and upper bounds for binary subspace codes for projective dimensions of at most $7$. We obtain several improvements of the bounds and perform two classifications of optimal subspace codes, which are unknown so far in the literature.","authors_text":"Daniel Heinlein, Sascha Kurz","cross_cats":["cs.IT","math.IT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-06T12:13:20Z","title":"Binary Subspace Codes in Small Ambient Spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.02219","kind":"arxiv","version":3},"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":"sha256:2c6ad0393bbdb1d96f9e14665fac23c0f4b926f1da0866be365469587e2cfff7","target":"record","created_at":"2026-05-18T00:04:37Z","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":"d3e568e70cc36b31c32825fc73714fd494717f8151ca26d8962592c50da6bbea","cross_cats_sorted":["cs.IT","math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-06T12:13:20Z","title_canon_sha256":"70efac4092383dba57e20458a05681d6ebd50da26799054ed0f5408815d384b4"},"schema_version":"1.0","source":{"id":"1804.02219","kind":"arxiv","version":3}},"canonical_sha256":"10ed7600e3891a067bd02dd86dde2a149abdba5cb65811f289e8ba89a7094b02","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"10ed7600e3891a067bd02dd86dde2a149abdba5cb65811f289e8ba89a7094b02","first_computed_at":"2026-05-18T00:04:37.455245Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:04:37.455245Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CWER88t9q3mqvPIhTWzynhAFOFnXps5KxKYcrXJpfzfxPL/MG0BuuFWK0ZLwuQDRHn90y+GcCFAyyDWSta24AQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:04:37.455743Z","signed_message":"canonical_sha256_bytes"},"source_id":"1804.02219","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2c6ad0393bbdb1d96f9e14665fac23c0f4b926f1da0866be365469587e2cfff7","sha256:0cd19234c051fa84978d43ca2147b4bc64e3dfb9dffe8746a202b44370698faa"],"state_sha256":"fdb2864afef4418253faffbc011ba204890cf0f96542963caa9e6c5aa8e61515"}