  Compressive sensing aims to recover a high-dimensional sparse signal from a
relatively small number of measurements. In this paper, a novel design of the
measurement matrix is proposed. The design is inspired by the construction of
generalized low-density parity-check codes, where the capacity-achieving
point-to-point codes serve as subcodes to robustly estimate the signal support.
In the case that each entry of the $n$-dimensional $k$-sparse signal lies in a
known discrete alphabet, the proposed scheme requires only $O(k \log n)$
measurements and arithmetic operations. In the case of arbitrary, possibly
continuous alphabet, an error propagation graph is proposed to characterize the
residual estimation error. With $O(k \log^2 n)$ measurements and computational
complexity, the reconstruction error can be made arbitrarily small with high
probability.
