We propose a new compressive sensing scheme, 
based on codes of graphs, that allows for joint design of sensing 
matrices and low complexity reconstruction algorithms. The 
compressive sensing matrices can be shown to offer asymptoti- 
cally optimal performance when used in combination with orthogonal
matching pursuit methods. Furthermore, we demonstrate an interesting connection between the binary restricted isometry property and higher weight enumerators of codes used for compressive sensing matrix design.
For more elaborate greedy reconstruction schemes, we 
propose a new family of list decoding and multiple-basis belief 
propagation algorithms. Our simulation results indicate that the 
proposed CS scheme offers good complexity-performance trade- 
offs for several classes of sparse signals.