Singular 0/1-Matrices, and the Hyperplanes Spanned by Random 0/1-Vectors

Abstract

Let ${P_s(d)}$ be the probability that a random 0/1-matrix of size $d \times d$ is singular, and let ${E(d)}$ be the expected number of 0/1-vectors in the linear subspace spanned by $d-1$ random independent 0/1-vectors. (So ${E(d)}$ is the expected number of cube vertices on a random affine hyperplane spanned by vertices of the cube.)We prove that bounds on ${P_s(d)}$ are equivalent to bounds on ${E(d)}$: \[{P_s(d)} = \bigg(2^{-d} {E(d)} + \frac{d^2}{2^{d+1}} \bigg) (1 + \so(1)). \] We also report on computational experiments pertaining to these numbers.