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${session.getAttribute("locale")}5Probabilistic lower bounds on maximal determinants of binary matrices
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n/2 be the ratio of D(n) to the Hadamard upper bound. Using the probabilistic method, we prove new lower bounds on D(n) and R(n) in terms of d = n - h, where h is the order of a Hadamard matrix and h is maximal subject to h ≤ n. For example, [forumal cannot be replicated]. By a recent result of Livinskyi, d²/h^{1/2} → 0 as n → 8, so the second bound is close to (πe/2)^{-d/2} for large n. Previous lower bounds tended to zero as n → ∞ with d fixed, except in the cases d ∈ {0, 1}. For d ≥ 2, our bounds are better for all sufficiently large n. If the Hadamard conjecture is true, then d ≤ 3, so the first bound above shows that R(n) is bounded below by a positive constant (πe/2)^{-3/2} > 0.1133.]]>Wed 11 Apr 2018 10:30:42 AEST]]>Note on best possible bounds for determinants of matrices close to the identity matrix
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Sat 24 Mar 2018 07:27:02 AEDT]]>