{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:ETFRO4RGCMOSTKK2N5WUQJGGCR","short_pith_number":"pith:ETFRO4RG","schema_version":"1.0","canonical_sha256":"24cb177226131d29a95a6f6d4824c6144553f5bd8ddeb8e3b5d3c0268d0d105f","source":{"kind":"arxiv","id":"1311.4925","version":1},"attestation_state":"computed","paper":{"title":"Asymptotic Improvement of the Gilbert-Varshamov Bound on the Size of Permutation Codes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.IT"],"primary_cat":"math.CO","authors_text":"Alexander Vardy, Jacques Verstraete, Michael Tait","submitted_at":"2013-11-20T00:33:16Z","abstract_excerpt":"Given positive integers $n$ and $d$, let $M(n,d)$ denote the maximum size of a permutation code of length $n$ and minimum Hamming distance $d$. The Gilbert-Varshamov bound asserts that $M(n,d) \\geq n!/V(n,d-1)$ where $V(n,d)$ is the volume of a Hamming sphere of radius $d$ in $\\S_n$.\n  Recently, Gao, Yang, and Ge showed that this bound can be improved by a factor $\\Omega(\\log n)$, when $d$ is fixed and $n \\to \\infty$. Herein, we consider the situation where the ratio $d/n$ is fixed and improve the Gilbert-Varshamov bound by a factor that is \\emph{linear in $n$}. That is, we show that if $d/n <"},"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":"1311.4925","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-11-20T00:33:16Z","cross_cats_sorted":["cs.IT","math.IT"],"title_canon_sha256":"1961615c6a5b895282d4bf1c083cf9ae1c8425cea3a195b9c4e29603f7b32c70","abstract_canon_sha256":"7aac96caa9a8b9655be4230910c6ec916af325ad96141b3890427ff069c944a8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:06:42.400037Z","signature_b64":"F/kUXnwQFc++Wo6+w8bu1S92NcNVPL7kFuPw0hM+s9Fe4NqZdaSi9LbXhVrExzxDVjeK9942eKoeBx/gSMS1BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"24cb177226131d29a95a6f6d4824c6144553f5bd8ddeb8e3b5d3c0268d0d105f","last_reissued_at":"2026-05-18T03:06:42.399582Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:06:42.399582Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotic Improvement of the Gilbert-Varshamov Bound on the Size of Permutation Codes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.IT"],"primary_cat":"math.CO","authors_text":"Alexander Vardy, Jacques Verstraete, Michael Tait","submitted_at":"2013-11-20T00:33:16Z","abstract_excerpt":"Given positive integers $n$ and $d$, let $M(n,d)$ denote the maximum size of a permutation code of length $n$ and minimum Hamming distance $d$. The Gilbert-Varshamov bound asserts that $M(n,d) \\geq n!/V(n,d-1)$ where $V(n,d)$ is the volume of a Hamming sphere of radius $d$ in $\\S_n$.\n  Recently, Gao, Yang, and Ge showed that this bound can be improved by a factor $\\Omega(\\log n)$, when $d$ is fixed and $n \\to \\infty$. Herein, we consider the situation where the ratio $d/n$ is fixed and improve the Gilbert-Varshamov bound by a factor that is \\emph{linear in $n$}. That is, we show that if $d/n <"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.4925","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":"1311.4925","created_at":"2026-05-18T03:06:42.399653+00:00"},{"alias_kind":"arxiv_version","alias_value":"1311.4925v1","created_at":"2026-05-18T03:06:42.399653+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.4925","created_at":"2026-05-18T03:06:42.399653+00:00"},{"alias_kind":"pith_short_12","alias_value":"ETFRO4RGCMOS","created_at":"2026-05-18T12:27:43.054852+00:00"},{"alias_kind":"pith_short_16","alias_value":"ETFRO4RGCMOSTKK2","created_at":"2026-05-18T12:27:43.054852+00:00"},{"alias_kind":"pith_short_8","alias_value":"ETFRO4RG","created_at":"2026-05-18T12:27:43.054852+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/ETFRO4RGCMOSTKK2N5WUQJGGCR","json":"https://pith.science/pith/ETFRO4RGCMOSTKK2N5WUQJGGCR.json","graph_json":"https://pith.science/api/pith-number/ETFRO4RGCMOSTKK2N5WUQJGGCR/graph.json","events_json":"https://pith.science/api/pith-number/ETFRO4RGCMOSTKK2N5WUQJGGCR/events.json","paper":"https://pith.science/paper/ETFRO4RG"},"agent_actions":{"view_html":"https://pith.science/pith/ETFRO4RGCMOSTKK2N5WUQJGGCR","download_json":"https://pith.science/pith/ETFRO4RGCMOSTKK2N5WUQJGGCR.json","view_paper":"https://pith.science/paper/ETFRO4RG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1311.4925&json=true","fetch_graph":"https://pith.science/api/pith-number/ETFRO4RGCMOSTKK2N5WUQJGGCR/graph.json","fetch_events":"https://pith.science/api/pith-number/ETFRO4RGCMOSTKK2N5WUQJGGCR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ETFRO4RGCMOSTKK2N5WUQJGGCR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ETFRO4RGCMOSTKK2N5WUQJGGCR/action/storage_attestation","attest_author":"https://pith.science/pith/ETFRO4RGCMOSTKK2N5WUQJGGCR/action/author_attestation","sign_citation":"https://pith.science/pith/ETFRO4RGCMOSTKK2N5WUQJGGCR/action/citation_signature","submit_replication":"https://pith.science/pith/ETFRO4RGCMOSTKK2N5WUQJGGCR/action/replication_record"}},"created_at":"2026-05-18T03:06:42.399653+00:00","updated_at":"2026-05-18T03:06:42.399653+00:00"}