This is a tale of two linear programming decoders, namely channel coding
linear programming decoding (CC-LPD) and compressed sensing linear programming
decoding (CS-LPD). So far, they have evolved quite independently. The aim of
the present paper is to show that there is a tight connection between, on the
one hand, CS-LPD based on a zero-one measurement matrix over the reals and, on
the other hand, CC-LPD of the binary linear code that is obtained by viewing
this measurement matrix as a binary parity-check matrix. This connection allows
one to translate performance guarantees from one setup to the other.