  Inspired by compressive sensing principles, we propose novel error control
coding techniques for communication systems. The information bits are encoded
in the support and the non-zero entries of a sparse signal. By selecting a
dictionary matrix with suitable dimensions, the codeword for transmission is
obtained by multiplying the dictionary matrix with the sparse signal.
Specifically, the codewords are obtained from the sparse linear combinations of
the columns of the dictionary matrix. At the decoder, we employ variations of
greedy sparse signal recovery algorithms. Using Gold code sequences and
mutually unbiased bases from quantum information theory as dictionary matrices,
we study the block error rate (BLER) performance of the proposed scheme in the
AWGN channel. Our results show that the proposed scheme has a comparable and
competitive performance with respect to the several widely used linear codes,
for very small to moderate block lengths. In addition, our coding scheme
extends straightforwardly to multi-user scenarios such as multiple access
channel, broadcast channel, and interference channel. In these multi-user
channels, if the users are grouped such that they have similar channel gains
and noise levels, the overall BLER performance of our proposed scheme will
coincide with an equivalent single-user scenario.
