  In this paper we discuss a novel data compression technique for binary
symmetric sources based on the cavity method over a Galois Field of order q
(GF(q)). We present a scheme of low complexity and near optimal empirical
performance. The compression step is based on a reduction of sparse low density
parity check codes over GF(q) and is done through the so called reinforced
belief-propagation equations. These reduced codes appear to have a non-trivial
geometrical modification of the space of codewords which makes such compression
computationally feasible. The computational complexity is O(d.n.q.log(q)) per
iteration, where d is the average degree of the check nodes and n is the number
of bits. For our code ensemble, decompression can be done in a time linear in
the code's length by a simple leaf-removal algorithm.
