  We consider a space of sparse Boolean matrices of size $n \times n$, which
have finite co-rank over $GF(2)$ with high probability. In particular, the
probability such a matrix has full rank, and is thus invertible, is a positive
constant with value about $0.2574$ for large $n$.
  The matrices arise as the vertex-edge incidence matrix of 1-out 3-uniform
hypergraphs The result that the null space is finite, can be contrasted with
results for the usual models of sparse Boolean matrices, based on the
vertex-edge incidence matrix of random $k$-uniform hypergraphs. For this latter
model, the expected co-rank is linear in the number of vertices $n$,
\cite{ACO}, \cite{CFP}.
  For fields of higher order, the co-rank is typically Poisson distributed.
