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 $R(n):=D(n)\mathrm{/}{n}^{n\mathrm{/}2}\ge {\kappa }_d>0$. Here $n^{n\mathrm{/}2}$ 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 $R(n) \ge (2/(\pi e))^{d/2}$forallsufficientlylarge$n$ (and conjecturally for all positive $n$). If the Hadamard conjecture is true, then $d\le 3$ and ${\kappa }_d\ge (2\mathrm{/}(\pi e){)}^{d\mathrm{/}2}>1\mathrm{/}9$.