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Using the probabilistic method, we prove new lower bounds on ${\\mathcal D}(n)$ and $\\mathcal R(n)$ in terms of $d = n-h$, where $h$ is the order of a Hadamard matrix and $h$ is maximal subject to $h \\le n$. For example, $\\mathcal R(n) > (\\pi e/2)^{-d/2}$ if $1 \\le d \\le 3$, and $\\mathcal R(n) > (\\pi e/2)^{-d/2}(1 - d^2(\\pi/(2h))^{1/2})$ if $d > 3$. By a recent result of Livinskyi, $d^2/h^{1/2} \\to 0$"},"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":"1501.06235","kind":"arxiv","version":7},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-26T02:19:57Z","cross_cats_sorted":[],"title_canon_sha256":"123bcde0a19716c4f8821b4f1635ce2c1228423656da2cdbaa28e83fc2ee0e5d","abstract_canon_sha256":"7da541c847b309399e1ec5dd623614d7832fa6ee34ff58a4480e261188bfe777"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:00:46.329496Z","signature_b64":"CZwD8yWr1aGBAeYhSCTRo/kmG4ysd5+euYxIkaA7ymxLEov78fzzy10fmc8mV2FTbcf/oJvhTvE+/WM5MT7yAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"36b21f35569647ec834b1ad62ecfcfb2c8fe82934e3e773f1252e63ef8756b9a","last_reissued_at":"2026-05-18T01:00:46.329092Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:00:46.329092Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Probabilistic lower bounds on maximal determinants of binary matrices","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":"2015-01-26T02:19:57Z","abstract_excerpt":"Let ${\\mathcal D}(n)$ be the maximal determinant for $n \\times n$ $\\{\\pm 1\\}$-matrices, and $\\mathcal R(n) = {\\mathcal D}(n)/n^{n/2}$ be the ratio of ${\\mathcal D}(n)$ to the Hadamard upper bound. Using the probabilistic method, we prove new lower bounds on ${\\mathcal D}(n)$ and $\\mathcal R(n)$ in terms of $d = n-h$, where $h$ is the order of a Hadamard matrix and $h$ is maximal subject to $h \\le n$. For example, $\\mathcal R(n) > (\\pi e/2)^{-d/2}$ if $1 \\le d \\le 3$, and $\\mathcal R(n) > (\\pi e/2)^{-d/2}(1 - d^2(\\pi/(2h))^{1/2})$ if $d > 3$. By a recent result of Livinskyi, $d^2/h^{1/2} \\to 0$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.06235","kind":"arxiv","version":7},"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":"1501.06235","created_at":"2026-05-18T01:00:46.329153+00:00"},{"alias_kind":"arxiv_version","alias_value":"1501.06235v7","created_at":"2026-05-18T01:00:46.329153+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.06235","created_at":"2026-05-18T01:00:46.329153+00:00"},{"alias_kind":"pith_short_12","alias_value":"G2ZB6NKWSZD6","created_at":"2026-05-18T12:29:22.688609+00:00"},{"alias_kind":"pith_short_16","alias_value":"G2ZB6NKWSZD6ZA2L","created_at":"2026-05-18T12:29:22.688609+00:00"},{"alias_kind":"pith_short_8","alias_value":"G2ZB6NKW","created_at":"2026-05-18T12:29:22.688609+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/G2ZB6NKWSZD6ZA2LDLLC5T6PWL","json":"https://pith.science/pith/G2ZB6NKWSZD6ZA2LDLLC5T6PWL.json","graph_json":"https://pith.science/api/pith-number/G2ZB6NKWSZD6ZA2LDLLC5T6PWL/graph.json","events_json":"https://pith.science/api/pith-number/G2ZB6NKWSZD6ZA2LDLLC5T6PWL/events.json","paper":"https://pith.science/paper/G2ZB6NKW"},"agent_actions":{"view_html":"https://pith.science/pith/G2ZB6NKWSZD6ZA2LDLLC5T6PWL","download_json":"https://pith.science/pith/G2ZB6NKWSZD6ZA2LDLLC5T6PWL.json","view_paper":"https://pith.science/paper/G2ZB6NKW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1501.06235&json=true","fetch_graph":"https://pith.science/api/pith-number/G2ZB6NKWSZD6ZA2LDLLC5T6PWL/graph.json","fetch_events":"https://pith.science/api/pith-number/G2ZB6NKWSZD6ZA2LDLLC5T6PWL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/G2ZB6NKWSZD6ZA2LDLLC5T6PWL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/G2ZB6NKWSZD6ZA2LDLLC5T6PWL/action/storage_attestation","attest_author":"https://pith.science/pith/G2ZB6NKWSZD6ZA2LDLLC5T6PWL/action/author_attestation","sign_citation":"https://pith.science/pith/G2ZB6NKWSZD6ZA2LDLLC5T6PWL/action/citation_signature","submit_replication":"https://pith.science/pith/G2ZB6NKWSZD6ZA2LDLLC5T6PWL/action/replication_record"}},"created_at":"2026-05-18T01:00:46.329153+00:00","updated_at":"2026-05-18T01:00:46.329153+00:00"}