{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:S4H45WGU77EJ6DSARBGHVPRQA3","short_pith_number":"pith:S4H45WGU","schema_version":"1.0","canonical_sha256":"970fced8d4ffc89f0e40884c7abe3006e1117422b8049ecc574b375f7cdbe2fd","source":{"kind":"arxiv","id":"1402.6817","version":6},"attestation_state":"computed","paper":{"title":"Lower bounds on maximal determinants of binary matrices via the probabilistic method","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Judy-anne H. Osborn, Richard P. Brent, Warren D. Smith","submitted_at":"2014-02-27T08:03:41Z","abstract_excerpt":"Let $D(n)$ be the maximal determinant for $n \\times n$ $\\{\\pm 1\\}$-matrices, and ${\\mathcal R}(n) = D(n)/n^{n/2}$ be the ratio of $D(n)$ to the Hadamard upper bound. We give several new lower bounds on ${\\mathcal R}(n)$ in terms of $d$, where $n = h+d$, $h$ is the order of a Hadamard matrix, and $h$ is maximal subject to $h \\le n$. A relatively simple bound is \\[{\\mathcal R}(n) \\ge \\left(\\frac{2}{\\pi e}\\right)^{d/2}\n  \\left(1 - d^2\\left(\\frac{\\pi}{2h}\\right)^{1/2}\\right)\n  \\;\\text{ for all }\\; n \\ge 1.\\] An asymptotically sharper bound is \\[{\\mathcal R}(n) \\ge \\left(\\frac{2}{\\pi e}\\right)^{d/2"},"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":"1402.6817","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-02-27T08:03:41Z","cross_cats_sorted":[],"title_canon_sha256":"a60a25aa809cf7e40162b7ba44183e68524245b83d119eb7ea5a6f43123c8501","abstract_canon_sha256":"a76bfacb266c8f610abb1541859ca08cf910c1ba144b1d6b14fd3a4bee56c40a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:01:29.535958Z","signature_b64":"v3uzdDKhTny2qnsWoMYCRvxk6nsfhNWvAo0CdW3E5yyKobQYmcj8bJsL7z23ulNu1XOHOMw9x46Jt+I1DNOTAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"970fced8d4ffc89f0e40884c7abe3006e1117422b8049ecc574b375f7cdbe2fd","last_reissued_at":"2026-05-18T01:01:29.535398Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:01:29.535398Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Lower bounds on maximal determinants of binary matrices via the probabilistic method","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Judy-anne H. Osborn, Richard P. Brent, Warren D. Smith","submitted_at":"2014-02-27T08:03:41Z","abstract_excerpt":"Let $D(n)$ be the maximal determinant for $n \\times n$ $\\{\\pm 1\\}$-matrices, and ${\\mathcal R}(n) = D(n)/n^{n/2}$ be the ratio of $D(n)$ to the Hadamard upper bound. We give several new lower bounds on ${\\mathcal R}(n)$ in terms of $d$, where $n = h+d$, $h$ is the order of a Hadamard matrix, and $h$ is maximal subject to $h \\le n$. A relatively simple bound is \\[{\\mathcal R}(n) \\ge \\left(\\frac{2}{\\pi e}\\right)^{d/2}\n  \\left(1 - d^2\\left(\\frac{\\pi}{2h}\\right)^{1/2}\\right)\n  \\;\\text{ for all }\\; n \\ge 1.\\] An asymptotically sharper bound is \\[{\\mathcal R}(n) \\ge \\left(\\frac{2}{\\pi e}\\right)^{d/2"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.6817","kind":"arxiv","version":6},"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":"1402.6817","created_at":"2026-05-18T01:01:29.535461+00:00"},{"alias_kind":"arxiv_version","alias_value":"1402.6817v6","created_at":"2026-05-18T01:01:29.535461+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1402.6817","created_at":"2026-05-18T01:01:29.535461+00:00"},{"alias_kind":"pith_short_12","alias_value":"S4H45WGU77EJ","created_at":"2026-05-18T12:28:46.137349+00:00"},{"alias_kind":"pith_short_16","alias_value":"S4H45WGU77EJ6DSA","created_at":"2026-05-18T12:28:46.137349+00:00"},{"alias_kind":"pith_short_8","alias_value":"S4H45WGU","created_at":"2026-05-18T12:28:46.137349+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/S4H45WGU77EJ6DSARBGHVPRQA3","json":"https://pith.science/pith/S4H45WGU77EJ6DSARBGHVPRQA3.json","graph_json":"https://pith.science/api/pith-number/S4H45WGU77EJ6DSARBGHVPRQA3/graph.json","events_json":"https://pith.science/api/pith-number/S4H45WGU77EJ6DSARBGHVPRQA3/events.json","paper":"https://pith.science/paper/S4H45WGU"},"agent_actions":{"view_html":"https://pith.science/pith/S4H45WGU77EJ6DSARBGHVPRQA3","download_json":"https://pith.science/pith/S4H45WGU77EJ6DSARBGHVPRQA3.json","view_paper":"https://pith.science/paper/S4H45WGU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1402.6817&json=true","fetch_graph":"https://pith.science/api/pith-number/S4H45WGU77EJ6DSARBGHVPRQA3/graph.json","fetch_events":"https://pith.science/api/pith-number/S4H45WGU77EJ6DSARBGHVPRQA3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/S4H45WGU77EJ6DSARBGHVPRQA3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/S4H45WGU77EJ6DSARBGHVPRQA3/action/storage_attestation","attest_author":"https://pith.science/pith/S4H45WGU77EJ6DSARBGHVPRQA3/action/author_attestation","sign_citation":"https://pith.science/pith/S4H45WGU77EJ6DSARBGHVPRQA3/action/citation_signature","submit_replication":"https://pith.science/pith/S4H45WGU77EJ6DSARBGHVPRQA3/action/replication_record"}},"created_at":"2026-05-18T01:01:29.535461+00:00","updated_at":"2026-05-18T01:01:29.535461+00:00"}