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We present a new family of error-correcting codes for the binary symmetric channel. These codes are designed to encode a sparse source, and are defined in terms of very sparse invertible matrices, in such a way that the decoder can treat the signal and the noise symmetrically. The decoding problem involves only very sparse matrices and sparse vectors, and so is a promising candidate for practical decoding.

It can be proved that these codes are ‘very good’, in that sequences of codes exist which, when optimally decoded, achieve information rates up to the Shannon limit.

We give experimental results using a free energy minimization algorithm and a belief propagation algorithm for decoding, demonstrating practical performance superior to that of both Bose-Chaudhury-Hocquenghem codes and Reed-Muller codes over a wide range of noise levels.