We show that the maximal determinant D(n) for n × n {±1}-Here n n/2 is the Hadamard upper bound, and κ d depends only on d := n − h, where h is the maximal order of a Hadamard matrix with h ≤ 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.We give various lower bounds on R(n) that depend only on d. For example, R(n) ≥ 0.07 (0.352) d > 3 −(d+3) . For any fixed d ≥ 0 we have R(n) ≥ (2/(πe)) d/2 for all sufficiently large n (and conjecturally for all positive n). If the Hadamard conjecture is true, then d ≤ 3 and κ d ≥ (2/(πe)) d/2 > 1/9.
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