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

Published online by Cambridge University Press:  07 April 2006

THOMAS VOIGT  and
GÜNTER M. ZIEGLER
THOMAS VOIGT
Affiliation:
Inst. Mathematics, MA 6-2, TU Berlin, D-10623 Berlin, Germany (e-mail: tvoigt@math.tu-berlin.de, ziegler@math.tu-berlin.de)
GÜNTER M. ZIEGLER
Affiliation:
Inst. Mathematics, MA 6-2, TU Berlin, D-10623 Berlin, Germany (e-mail: tvoigt@math.tu-berlin.de, ziegler@math.tu-berlin.de)
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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.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 15 , Issue 3 , May 2006 , pp. 463 - 471
Copyright
2006 Cambridge University Press

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