Good Codes Based on Very Sparse Matrices

David MacKay , Cavendish Laboratory, Cambridge University
Radford M. Neal , Depts. of Statistics and Computer Science, University of Toronto

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.

In C. Boyd (editor) Cryptography and Coding: 5th IMA Conference, Lecture Notes in Computer Science No. 1025, pp. 100-111. Springer-Verlag.