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 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 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) are