Singular 0/1-matrices, and the hyperplanes spanned by random 0/1-vectors

Details Singular 0/1-matrices, and the hyperplanes spanned by random 0/1-vectors Singular 0/1-matrices, and the hyperplanes spanned by random 0/1-vectors

2003-08-06

arxiv.org/abs/math/0308050v3

Combinatorics, Probability & Computing, 15:463-471, 2006

Mathematics

Combinatorics

9 pages

Scientific paper

Let $P(d)$ 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 $P(d)$ are equivalent to bounds on $E(d)$: \[ P(d) = (2^{-d} E(d) + \frac{d^2}{2^{d+1}}) (1 + o(1)). \] We also report about computational experiments pertaining to these numbers.

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