{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:ZC3LLLDP6R4PCTESW6NFSGDJCF","short_pith_number":"pith:ZC3LLLDP","schema_version":"1.0","canonical_sha256":"c8b6b5ac6ff478f14c92b79a591869117c957f7538c84c65cc0a83ec7b00a00e","source":{"kind":"arxiv","id":"1808.09301","version":2},"attestation_state":"computed","paper":{"title":"New covering codes of radius $R$, codimension $tR$ and $tR+\\frac{R}{2}$, and saturating sets in projective spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alexander A. Davydov, Fernanda Pambianco, Stefano Marcugini","submitted_at":"2018-08-28T13:53:46Z","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{"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1808.09301","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-08-28T13:53:46Z","cross_cats_sorted":[],"title_canon_sha256":"8e5b72c99beca3cc7d04d3b4e10dc71c34f33026af21e0686cbd6fc13f38c04e","abstract_canon_sha256":"8d8c69a48374dd5b3d5c47ed46d5e44411e39a29bef6fc84c6531ff722eb4e66"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:51:09.633980Z","signature_b64":"C1Smbcy6KBmOjUS8vxV+vEUx39u+8dlbpAGiDJ9oxwyX6doESzgcDc1jjAm8dSCVLscvQYcho4RTYpX5Jan3Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c8b6b5ac6ff478f14c92b79a591869117c957f7538c84c65cc0a83ec7b00a00e","last_reissued_at":"2026-05-17T23:51:09.633483Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:51:09.633483Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"New covering codes of radius $R$, codimension $tR$ and $tR+\\frac{R}{2}$, and saturating sets in projective spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alexander A. Davydov, Fernanda Pambianco, Stefano Marcugini","submitted_at":"2018-08-28T13:53:46Z","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{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.09301","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1808.09301","created_at":"2026-05-17T23:51:09.633558+00:00"},{"alias_kind":"arxiv_version","alias_value":"1808.09301v2","created_at":"2026-05-17T23:51:09.633558+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.09301","created_at":"2026-05-17T23:51:09.633558+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZC3LLLDP6R4P","created_at":"2026-05-18T12:33:04.347982+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZC3LLLDP6R4PCTES","created_at":"2026-05-18T12:33:04.347982+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZC3LLLDP","created_at":"2026-05-18T12:33:04.347982+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ZC3LLLDP6R4PCTESW6NFSGDJCF","json":"https://pith.science/pith/ZC3LLLDP6R4PCTESW6NFSGDJCF.json","graph_json":"https://pith.science/api/pith-number/ZC3LLLDP6R4PCTESW6NFSGDJCF/graph.json","events_json":"https://pith.science/api/pith-number/ZC3LLLDP6R4PCTESW6NFSGDJCF/events.json","paper":"https://pith.science/paper/ZC3LLLDP"},"agent_actions":{"view_html":"https://pith.science/pith/ZC3LLLDP6R4PCTESW6NFSGDJCF","download_json":"https://pith.science/pith/ZC3LLLDP6R4PCTESW6NFSGDJCF.json","view_paper":"https://pith.science/paper/ZC3LLLDP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1808.09301&json=true","fetch_graph":"https://pith.science/api/pith-number/ZC3LLLDP6R4PCTESW6NFSGDJCF/graph.json","fetch_events":"https://pith.science/api/pith-number/ZC3LLLDP6R4PCTESW6NFSGDJCF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZC3LLLDP6R4PCTESW6NFSGDJCF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZC3LLLDP6R4PCTESW6NFSGDJCF/action/storage_attestation","attest_author":"https://pith.science/pith/ZC3LLLDP6R4PCTESW6NFSGDJCF/action/author_attestation","sign_citation":"https://pith.science/pith/ZC3LLLDP6R4PCTESW6NFSGDJCF/action/citation_signature","submit_replication":"https://pith.science/pith/ZC3LLLDP6R4PCTESW6NFSGDJCF/action/replication_record"}},"created_at":"2026-05-17T23:51:09.633558+00:00","updated_at":"2026-05-17T23:51:09.633558+00:00"}