{"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"}