Necessary and Sufficient Conditions for Recovery of Sparse Signals over Finite Fields

We consider a compressed sensing (CS) framework over finite fields. We derive sufficient and necessary conditions for recovery of sparse signals in terms of the ambient dimension of the signal space, the sparsity of the signal, the number of measurements, and the field size. We show that the sufficient condition coincides with the necessary condition if the sensing matrix is sufficiently dense while both the length of the signal and the field size grow to infinity. One of the interesting conclusions includes that unless the signal is very sparse, the sensing matrix does not have to be dense to have the upper bound coincide with the lower bound.