(1, -1)-Matrices with Near-Extremal Properties

We prove the existence of $(-1,1)$-matrices with near-extremal properties. In particular, we find matrices having either small inner products between all pairs of distinct rows or having determinants approaching Hadamard's bound. Our approach is to study the $m \times n$ submatrices of a Hadamard matrix of order $k$. We use both the probabilistic method to obtain information about randomly selected submatrices and also combinatorial bounds which hold for all submatrices. Our existence results depend upon the gaps between successive primes. Beyond the applications presented in this paper, the study of randomly selected submatrices of combinatorial designs is of intrinsic interest.