Lower bounds on maximal determinants of +-1 matrices via the probabilistic method

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 $d > 1$. 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 sufficiently large $n$ (and conjecturally for all positive $n$). If the Hadamard conjecture is true, then $d \le 3$ and $\kappa_d \ge (2/(\pi e))^{d/2} > 1/9$.