{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2002:BMKSIARZRFEEY5Z45D32FNK73H","short_pith_number":"pith:BMKSIARZ","schema_version":"1.0","canonical_sha256":"0b1524023989484c773ce8f7a2b55fd9f981ec7f9509936bb898572e39d7138c","source":{"kind":"arxiv","id":"math/0205303","version":1},"attestation_state":"computed","paper":{"title":"On Asymmetric Coverings and Covering Numbers","license":"","headline":"","cross_cats":["cs.IT","math.IT"],"primary_cat":"math.CO","authors_text":"David Applegate, E. M. Rains, N. J. A. Sloane","submitted_at":"2002-05-28T21:50:03Z","abstract_excerpt":"An asymmetric covering D(n,R) is a collection of special subsets S of an n-set such that every subset T of the n-set is contained in at least one special S with |S| - |T| <= R. In this paper we compute the smallest size of any D(n,1) for n <= 8. We also investigate ``continuous'' and ``banded'' versions of the problem. The latter involves the classical covering numbers C(n,k,k-1), and we determine the following new values: C(10,5,4) = 51, C(11,7,6,) =84, C(12,8,7) = 126, C(13,9,8)= 185 and C(14,10,9) = 259. We also find the number of nonisomorphic minimal covering designs in several cases."},"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":"math/0205303","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"2002-05-28T21:50:03Z","cross_cats_sorted":["cs.IT","math.IT"],"title_canon_sha256":"b2cb4691b3aaf5f49a8e88767e99d10e24f1230da6a81484aa216d4904f36f6a","abstract_canon_sha256":"56a47857a2cdee9b53efecdc49053e5c6bdd1f2f9a9c8b909c35a43de8c75d48"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:42:37.779921Z","signature_b64":"vqbXNOSsHvjWsgEUmthDSO1yuarELAH39cNMQ91vMhN69aojUY2eSBOUOVswzXt1yOzMKUcozvtSdGdP4PThAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0b1524023989484c773ce8f7a2b55fd9f981ec7f9509936bb898572e39d7138c","last_reissued_at":"2026-05-18T02:42:37.779216Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:42:37.779216Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Asymmetric Coverings and Covering Numbers","license":"","headline":"","cross_cats":["cs.IT","math.IT"],"primary_cat":"math.CO","authors_text":"David Applegate, E. M. Rains, N. J. A. Sloane","submitted_at":"2002-05-28T21:50:03Z","abstract_excerpt":"An asymmetric covering D(n,R) is a collection of special subsets S of an n-set such that every subset T of the n-set is contained in at least one special S with |S| - |T| <= R. In this paper we compute the smallest size of any D(n,1) for n <= 8. We also investigate ``continuous'' and ``banded'' versions of the problem. The latter involves the classical covering numbers C(n,k,k-1), and we determine the following new values: C(10,5,4) = 51, C(11,7,6,) =84, C(12,8,7) = 126, C(13,9,8)= 185 and C(14,10,9) = 259. We also find the number of nonisomorphic minimal covering designs in several cases."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0205303","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"math/0205303","created_at":"2026-05-18T02:42:37.779328+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0205303v1","created_at":"2026-05-18T02:42:37.779328+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0205303","created_at":"2026-05-18T02:42:37.779328+00:00"},{"alias_kind":"pith_short_12","alias_value":"BMKSIARZRFEE","created_at":"2026-05-18T12:25:50.845339+00:00"},{"alias_kind":"pith_short_16","alias_value":"BMKSIARZRFEEY5Z4","created_at":"2026-05-18T12:25:50.845339+00:00"},{"alias_kind":"pith_short_8","alias_value":"BMKSIARZ","created_at":"2026-05-18T12:25:50.845339+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/BMKSIARZRFEEY5Z45D32FNK73H","json":"https://pith.science/pith/BMKSIARZRFEEY5Z45D32FNK73H.json","graph_json":"https://pith.science/api/pith-number/BMKSIARZRFEEY5Z45D32FNK73H/graph.json","events_json":"https://pith.science/api/pith-number/BMKSIARZRFEEY5Z45D32FNK73H/events.json","paper":"https://pith.science/paper/BMKSIARZ"},"agent_actions":{"view_html":"https://pith.science/pith/BMKSIARZRFEEY5Z45D32FNK73H","download_json":"https://pith.science/pith/BMKSIARZRFEEY5Z45D32FNK73H.json","view_paper":"https://pith.science/paper/BMKSIARZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0205303&json=true","fetch_graph":"https://pith.science/api/pith-number/BMKSIARZRFEEY5Z45D32FNK73H/graph.json","fetch_events":"https://pith.science/api/pith-number/BMKSIARZRFEEY5Z45D32FNK73H/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BMKSIARZRFEEY5Z45D32FNK73H/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BMKSIARZRFEEY5Z45D32FNK73H/action/storage_attestation","attest_author":"https://pith.science/pith/BMKSIARZRFEEY5Z45D32FNK73H/action/author_attestation","sign_citation":"https://pith.science/pith/BMKSIARZRFEEY5Z45D32FNK73H/action/citation_signature","submit_replication":"https://pith.science/pith/BMKSIARZRFEEY5Z45D32FNK73H/action/replication_record"}},"created_at":"2026-05-18T02:42:37.779328+00:00","updated_at":"2026-05-18T02:42:37.779328+00:00"}