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arXiv:cs/0602087 · IEEE Information Theory Workshop (ITW)

Bounds on the Threshold of Linear Programming Decoding

Pascal O. Vontobel, R. Koetter

Published May 25, 2006• 16 citations
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Abstract

Whereas many results are known about thresholds for ensembles of low-density parity-check codes under message-passing iterative decoding, this is not the case for linear programming decoding. Towards closing this knowledge gap, this paper presents some bounds on the thresholds of low-density parity-check code ensembles under linear programming decoding.

DOI
10.1109/itw.2006.1633805

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Whereas many results are known about thresholds for ensembles of low-density parity-check codes under message-passing iterative decoding, this is not the case for linear programming decoding. Towards closing this knowledge gap, this paper presents some bounds on the thresholds of low-density parity-check code ensembles under linear programming decoding.
Linear programming (LP) decoding is an alternative to iterative algorithms for decoding low density parity check (LDPC) codes. Although the practical performance of LP decoding is comparable to message-passing decoding, a significant advantage is its relative amenability to nonasymptotic analysis. Moreover, there turn out to be a number of important theoretical connections between the LP decoding and standard forms of iterative decoding. These connections allow theoretical insight from the LP decoding perspective to be transferred to iterative decoding algorithms. These advantages encouraged many researchers to work in this recent decoding technique for LDPC codes. In this paper, LP decoding for LDPC code is extensively reviewed and is discussed in different segmented areas.
Linear programming decoding for low-density parity check codes (and related domains such as compressed sensing) has received increased attention over recent years because of its practical performance --coming close to that of iterative decoding algorithms--- and its amenability to finite-blocklength analysis. Several works starting with the work of Feldman et al. showed how to analyze LP decoding using properties of expander graphs. This line of analysis works for only low error rates, about a couple of orders of magnitude lower than the empirically observed performance. It is possible to do better for the case of random noise, as shown by Daskalakis et al. and Koetter and Vontobel. Building on work of Koetter and Vontobel, we obtain a novel understanding of LP decoding, which allows us to establish a 0.05-fraction of correctable errors for rate-1/2 codes; this comes very close to the performance of iterative decoders and is significantly higher than the best previously noted correctable bit error rate for LP decoding. Unlike other techniques, our analysis directly works with the primal linear program and exploits an explicit connection between LP decoding and message passing algorithms.
This work considers the behavior of the degree distributions of capacity-approaching low-density parity-check (LDPC) code ensembles via linear programming (LP) bounds. These LP bounds are information-theoretic, and they apply to finite-length LDPC codes and to the asymptotic case of an infinite block length. Analytical solutions of these bounds are given in closed form, and the bounds are compared for the BEC with some specific degree distributions of capacity-achieving sequences of LDPC code ensembles. These LP bounds are shown to be informative and are easy to calculate. Due to space limitations, the derivation of the LP bounds is outlined, and the reader is referred to the full paper version for complete proofs and further discussions.