  Motivated by applications in unsourced random access, this paper develops a
novel scheme for the problem of compressed sensing of binary signals. In this
problem, the goal is to design a sensing matrix $A$ and a recovery algorithm,
such that the sparse binary vector $\mathbf{x}$ can be recovered reliably from
the measurements $\mathbf{y}=A\mathbf{x}+\sigma\mathbf{z}$, where $\mathbf{z}$
is additive white Gaussian noise. We propose to design $A$ as a parity check
matrix of a low-density parity-check code (LDPC), and to recover $\mathbf{x}$
from the measurements $\mathbf{y}$ using a Markov chain Monte Carlo algorithm,
which runs relatively fast due to the sparse structure of $A$. The performance
of our scheme is comparable to state-of-the-art schemes, which use dense
sensing matrices, while enjoying the advantages of using a sparse sensing
matrix.
