Coding Theory and applications

Lowering the error floor of Gallager codes: a statistical-mechanical view

The problem of error correction for Gallager's low-density parity-check codes is notoriously equivalent to that of computing marginal Boltzmann probabilities for an Ising-like model with multispin interactions in a non-uniform magnetic field. Since the graph of interactions is locally a tree, the solution is very well approximated by a generalized mean-field (Bethe--Peierls) approximation. Belief propagation (BP) and similar iterative algorithms are an efficient way to perform the calculation, but they sometimes fail to converge, or converge to non-codewords, giving rise to a non-negligible residual error probability (error floor). On the other hand, provably-convergent algorithms are far too complex to be implemented in a real decoder. In this work we consider the application of the probability-damping technique, which can be regarded either as a variant of BP, from which it retains the property of low complexity, or as an approximation of a provably-convergent algorithm, from which it is expected to inherit better convergence properties. We investigate the algorithm behaviour on a real instance of Gallager code, and compare the results with state-of-the-art algorithms.

Thu, 18/12/2014 - 14:30
Marco Pretti (Politecnico Torino)
Sala Affrescata, HuGeF, Molecular Biology Center (MBC), Via Nizza 52

RBP algorithm for lossy compression in reduced, ultrasparse GF(q) codes

gf-rbp.tgz24.36 KB

This code implements a novel data compression technique for binary symmetric sources based on the cavity method over GF(q), the Galois Field of order q. We present a scheme of low complexity and near-optimal empirical performance. The compression step is based on a reduction of a sparse low-density parity-check code over GF(q) and is done through the so-called reinforced belief-propagation equations. These reduced codes appear to have a nontrivial geometrical modification of the space of codewords, which makes such compression computationally feasible.

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