Lower bounds on maximal determinants of binary matrices via the probabilistic method
نویسنده
چکیده
Let D(n) be the maximal determinant for n × n {±1}-matrices, and R(n) = D(n)/n be the ratio of D(n) to the Hadamard upper bound. We give several new lower bounds on 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 ≤ n. A relatively simple bound is R(n) ≥ ( 2 πe )d/2( 1− d ( π 2h )1/2) for all n ≥ 1. An asymptotically sharper bound is R(n) ≥ ( 2 πe )d/2 exp ( d ( π 2h )1/2 + O ( d h2/3 )) . We also show that R(n) ≥ ( 2 πe )d/2 if n ≥ n0 and n0 is sufficiently large, the threshold n0 being independent of d, or for all n ≥ 1 if 0 ≤ d ≤ 3 (which would follow from the Hadamard conjecture). The proofs depend on the probabilistic method, and generalise previous results that were restricted to the cases d = 0 and d = 1.
منابع مشابه
Probabilistic lower bounds on maximal determinants of binary matrices
Let D(n) be the maximal determinant for n × n {±1}-matrices, and R(n) = D(n)/n 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 the distance d to the nearest (smaller) Hadamard matrix, defined by d = n − h, where h is the order of a Hadamard matrix and h is maximal subject to h ≤ n. The lower bounds on R(n) ...
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