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Australian Communications Theory Workshop

Generalized Belief Propagation to break trapping sets in LDPC codes

Jean-Christophe Sibel, Sylvain Reynal, David Declercq

Published Feb 1, 2014• 1 citation
Publisher

Abstract

In this paper, we focus on the Generalized Belief Propagation (GBP) algorithm to solve trapping sets in Low-Density Parity-Check (LDPC) codes. Trapping sets are topological structures in Tanner graphs of LDPC codes that are not correctly decoded by Belief Propagation (BP), leading to exhibit an error-floor in the Bit-Error Rate (BER). Stemming from statistical physics of spin glasses, GBP consists in passing messages between clusters of Tanner graph nodes in another graph called the region-graph. Here, we introduce a specific clustering of nodes, based on a novel local loopfree principle, that breaks trapping sets such that the resulting region-graph is locally loopfree. We then construct a hybrid decoder made of BP and GBP that proves to be a powerful decoder as it clearly improves the BER and defeats the error-floor.

DOI
10.1109/ausctw.2014.6766441

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In this paper, we focus on the Generalized Belief Propagation (GBP) algorithm to solve trapping sets in Low-Density Parity-Check (LDPC) codes. Trapping sets are topological structures in Tanner graphs of LDPC codes that are not correctly decoded by Belief Propagation (BP), leading to exhibit an error-floor in the Bit-Error Rate (BER). Stemming from statistical physics of spin glasses, GBP consists in passing messages between clusters of Tanner graph nodes in another graph called the region-graph. Here, we introduce a specific clustering of nodes, based on a novel local loopfree principle, that breaks trapping sets such that the resulting region-graph is locally loopfree. We then construct a hybrid decoder made of BP and GBP that proves to be a powerful decoder as it clearly improves the BER and defeats the error-floor.
Abstract—In this paper, we focus on the Generalized Belief Propagation (GBP) algorithm to solve trapping sets in Low-Density Parity-Check (LDPC) codes. Trapping sets are topo-logical structures in Tanner graphs of LDPC codes that are not correctly decoded by Belief Propagation (BP), leading to exhibit an error-floor in the Bit-Error Rate (BER). Stemming from statistical physics of spin glasses, GBP consists in passing messages between clusters of Tanner graph nodes in another graph called the region-graph. Here, we introduce a specific clustering of nodes, based on a novel local loopfree principle, that breaks trapping sets such that the resulting region-graph is locally loopfree. We then construct a hybrid decoder made of BP and GBP that proves to be a powerful decoder as it clearly improves the BER and defeats the error-floor. Index Terms—LDPC codes, Generalized Belief Propagation, trapping sets, error-floor, local clustering
The Belief Propagation (BP) is an inference algorithm used to estimate marginal probability distributions for any Markov Random Field (MRF). In the realm of Low-Density Parity-Check (LDPC) codes that can be represented by MRF called Tanner graphs, the BP is used as a decoding algorithm to estimate the states of bits sent through a noisy channel. Known to be optimal when the Tanner graph is a tree, the BP suffers from suboptimality when the Tanner graph has a loop-like topology. Furthermore, combinations of loops, namely the trapping sets, are particularly harmful for the decoding. To circumvent this problem were proposed other algorithms, like the Generalized Belief Propagation (GBP) that comes from statistical physics. This algorithm allows to absorb topological structures inside new nodes called regions. An advantage is that the resulting graph, the region graph, is not unique then according to its construction this region graph is a media for the GBP that can provide more accurate estimates than the BP. In this paper, we propose novel constructions of the region graph for the famous Tanner code of length N = 155 by making use of the trapping sets as basis for the regions.
Generalized belief propagation (GBP) is known to be a well-suited technique for approximate inference problems in loopy factor graphs. It can absorb problematic subgraphs inside regions to reduce their influence on the inference. However, the choice of regions to be used in GBP remains a delicate issue. This paper proposes an approach to create specific regions when dealing with Low-Density Parity-Check (LDPC) codes. We split trapping sets, known to degrade the decoding performance, to make GBP locally optimal. Experiments show that GBP can then perform better than BP, especially in the error-floor region.
Numerous inference problems in statistical physics, computer vision or error-correcting coding theory consist in approximating the marginal probability distributions on Markov Random Fields (MRF). The Belief Propagation (BP) is an accurate solution that is optimal if the MRF is loop free and suboptimal otherwise. In the context of error-correcting coding theory, any Low-Density Parity-Check (LDPC) code has a graphical representation, the Tanner graph, which is a particular MRF. It is used as a media for the BP algorithm to correct the bits, damaged by a noisy channel, by estimating their probability distributions. Though loops and combination thereof in the Tanner graph prevent the BP from being optimal, especially harmful topological structures called the trapping-sets. The BP has been extended to the Generalized Belief Propagation (GBP). This message-passing algorithm runs on a non unique mapping of the Tanner graph, namely the regiongraph, such that its nodes are gatherings of the Tanner graph nodes. Then it appears the possibility to decrease the loops effect, making the GBP more accurate than the BP. In this article, we expose a novel region graph construction suited to the Tanner code, an LDPC code whose Tanner graph is entirely covered by trapping-sets. Furthermore, we investigate the dynamic behavior of the GBP compared with that of the BP to understand its evolution in terms of the Signal-to-Noise Ratio (SNR). To this end we make use of classical estimators and we introduce a new one called the hyperspheres method.
This thesis addresses the problem of inference in factor graphs, especially the LDPC codes, almost solved by message-passing algorithms. In particular, the Belief Propagation algorithm (BP) is investigated as a particular message-passing algorithm whose suboptimality is discussed in the case where the factor graph has a loop-like topology. From the equivalence between the BP and the Bethe approximation in statistical physics that is generalized to the region-based approximation, is detailed the Generalized Belief Propagation algorithm (GBP), a message-passing algorithm between clusters of the factor graph. It is experimentally shown to surpass the BP in the cases where the clustering deals with the harmful topological structures that prevents the BP from rightly decoding any LDPC code, namely the trapping sets. We do not only confront the BP and the GBP algorithms according to their performance from the point of view of the channel coding with the error-rate, but also according to their dynamical behaviors for non-trivial error-events for which both algorithms can exhibit chaotic beahviors. By means of classical and original dynamical quantifiers, it is shown that the GBP algorithm can overcome the BP algorithm.
Quantum low-density parity check (QLDPC) codes are promising in realization of scalable, fault tolerant quantum memory for computation. Many of the QLDPC codes constructions suffer from unavoidable short cycles in their Tanner graph which degrade the decoding performance of the belief propagation (BP) algorithm. In this paper, we propose a syndrome based generalized belief propagation (GBP) algorithm for decoding of quantum LDPC codes and analyze how the proposed algorithm escapes from short cycle trapping sets effectively compared to the BP algorithm. Simulation results show improved decoding performance of the GBP algorithm over BP for the dual containing Calderbank, Shor and Steane (CSS) codes when cycles of length 4 are considered in the region based approach.