{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:ZC3LLLDP6R4PCTESW6NFSGDJCF","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":"8d8c69a48374dd5b3d5c47ed46d5e44411e39a29bef6fc84c6531ff722eb4e66","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-08-28T13:53:46Z","title_canon_sha256":"8e5b72c99beca3cc7d04d3b4e10dc71c34f33026af21e0686cbd6fc13f38c04e"},"schema_version":"1.0","source":{"id":"1808.09301","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1808.09301","created_at":"2026-05-17T23:51:09Z"},{"alias_kind":"arxiv_version","alias_value":"1808.09301v2","created_at":"2026-05-17T23:51:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.09301","created_at":"2026-05-17T23:51:09Z"},{"alias_kind":"pith_short_12","alias_value":"ZC3LLLDP6R4P","created_at":"2026-05-18T12:33:04Z"},{"alias_kind":"pith_short_16","alias_value":"ZC3LLLDP6R4PCTES","created_at":"2026-05-18T12:33:04Z"},{"alias_kind":"pith_short_8","alias_value":"ZC3LLLDP","created_at":"2026-05-18T12:33:04Z"}],"graph_snapshots":[{"event_id":"sha256:54ee8c4c710befe8500912aad9f4ef8ea4a491e9be97706ba1a13e33f8be8820","target":"graph","created_at":"2026-05-17T23:51:09Z","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":"The length function $\\ell_q(r,R)$ is the smallest length of a $ q $-ary linear code of codimension $r$ and covering radius $R$. In this work we obtain new constructive upper bounds on $\\ell_q(r,R)$ for all $R\\ge4$, $r=tR$, $t\\ge2$, and also for all even $R\\ge2$, $r=tR+\\frac{R}{2}$, $t\\ge1$. The new bounds are provided by infinite families of new covering codes with fixed $R$ and increasing codimension. The new bounds improve upon the known ones. We propose a general regular construction (called ``Line+Ovals'') of a minimal $\\rho$-saturating $((\\rho+1)q+1)$-set in the projective space $\\mathrm{","authors_text":"Alexander A. Davydov, Fernanda Pambianco, Stefano Marcugini","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-08-28T13:53:46Z","title":"New covering codes of radius $R$, codimension $tR$ and $tR+\\frac{R}{2}$, and saturating sets in projective spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.09301","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":"sha256:3b634d6462c5c4f4fdd5d7218245644f3794df9b412a16805c5154ee48d63789","target":"record","created_at":"2026-05-17T23:51:09Z","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":"8d8c69a48374dd5b3d5c47ed46d5e44411e39a29bef6fc84c6531ff722eb4e66","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-08-28T13:53:46Z","title_canon_sha256":"8e5b72c99beca3cc7d04d3b4e10dc71c34f33026af21e0686cbd6fc13f38c04e"},"schema_version":"1.0","source":{"id":"1808.09301","kind":"arxiv","version":2}},"canonical_sha256":"c8b6b5ac6ff478f14c92b79a591869117c957f7538c84c65cc0a83ec7b00a00e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c8b6b5ac6ff478f14c92b79a591869117c957f7538c84c65cc0a83ec7b00a00e","first_computed_at":"2026-05-17T23:51:09.633483Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:51:09.633483Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"C1Smbcy6KBmOjUS8vxV+vEUx39u+8dlbpAGiDJ9oxwyX6doESzgcDc1jjAm8dSCVLscvQYcho4RTYpX5Jan3Cw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:51:09.633980Z","signed_message":"canonical_sha256_bytes"},"source_id":"1808.09301","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3b634d6462c5c4f4fdd5d7218245644f3794df9b412a16805c5154ee48d63789","sha256:54ee8c4c710befe8500912aad9f4ef8ea4a491e9be97706ba1a13e33f8be8820"],"state_sha256":"0fb03518838d8952f919544701cec6d6a4c5f58290f23acf0e181e1ca500301a"}