{"paper":{"title":"Lower bounds on maximal determinants of +-1 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":"2012-11-14T09:31:09Z","abstract_excerpt":"We show that the maximal determinant D(n) for $n \\times n$ ${\\pm 1}$-matrices satisfies $R(n) := D(n)/n^{n/2} \\ge \\kappa_d > 0$. Here $n^{n/2}$ is the Hadamard upper bound, and $\\kappa_d$ depends only on $d := n-h$, where $h$ is the maximal order of a Hadamard matrix with $h \\le n$. Previous lower bounds on R(n) depend on both $d$ and $n$. Our bounds are improvements, for all sufficiently large $n$, if $d > 1$.\n  We give various lower bounds on R(n) that depend only on $d$. For example, $R(n) \\ge 0.07 (0.352)^d > 3^{-(d+3)}$. For any fixed $d \\ge 0$ we have $R(n) \\ge (2/(\\pi e))^{d/2}$ for all"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.3248","kind":"arxiv","version":3},"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"}