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

Abstract

The error correction for Gallager's low-density parity-check codes is a problem famously 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 method 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.