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IEEE International Symposium on Information Theory (ISIT)

On LDPC code ensembles with generalized constraints

Yanfang Liu, Pablo M. Olmos, Tobias Koch

Published Jun 1, 2017• 1 citation
Publisher

Abstract

In this paper, we analyze the tradeoff between coding rate and asymptotic performance of a class of generalized low-density parity-check (GLDPC) codes constructed by including a certain fraction of generalized constraint (GC) nodes in the graph. The rate of the GLDPC ensemble is bounded using classical results on linear block codes, namely Hamming bound and Varshamov bound. We also study the impact of the decoding method used at GC nodes. To incorporate both bounded-distance (BD) and Maximum Likelihood (ML) decoding at GC nodes into our analysis without having to resort on multi-edge type of degree distributions (DDs), we propose the probabilistic peeling decoder (P-PD) algorithm, which models the decoding step at every GC node as an instance of a Bernoulli random variable with a success probability that depends on the GC block code and its decoding algorithm. The P-PD asymptotic performance over the BEC can be efficiently predicted using standard techniques for LDPC codes such as density evolution (DE) or the differential equation method. Furthermore, for a class of GLDPC ensembles, we demonstrate that the simulated P-PD performance accurately predicts the actual performance of the GLPDC code. We illustrate our analysis for GLDPC code ensembles using (2, 6) and (2,15) base DDs. In all cases, we show that a large fraction of GC nodes is required to reduce the original gap to capacity.

DOI
10.1109/isit.2017.8006552

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In this paper, we analyze the tradeoff between coding rate and asymptotic performance of a class of generalized low-density parity-check (GLDPC) codes constructed by including a certain fraction of generalized constraint (GC) nodes in the graph. The rate of the GLDPC ensemble is bounded using classical results on linear block codes, namely Hamming bound and Varshamov bound. We also study the impact of the decoding method used at GC nodes. To incorporate both bounded-distance (BD) and Maximum Likelihood (ML) decoding at GC nodes into our analysis without having to resort on multi-edge type of degree distributions (DDs), we propose the probabilistic peeling decoder (P-PD) algorithm, which models the decoding step at every GC node as an instance of a Bernoulli random variable with a success probability that depends on the GC block code and its decoding algorithm. The P-PD asymptotic performance over the BEC can be efficiently predicted using standard techniques for LDPC codes such as density evolution (DE) or the differential equation method. Furthermore, for a class of GLDPC ensembles, we demonstrate that the simulated P-PD performance accurately predicts the actual performance of the GLPDC code. We illustrate our analysis for GLDPC code ensembles using (2, 6) and (2,15) base DDs. In all cases, we show that a large fraction of GC nodes is required to reduce the original gap to capacity.
In this paper, we analyze the tradeoff between coding rate and asymptotic performance of a class of generalized low-density parity-check (GLDPC) codes constructed by including a certain fraction of generalized constraint (GC) nodes in the graph. The rate of the GLDPC ensemble is bounded using classical results on linear block codes, namely, Hamming bound and Varshamov bound. We also study the impact of the decoding method used at GC nodes. To incorporate both bounded-distance (BD) and maximum likelihood (ML) decoding at GC nodes into our analysis without resorting on multi-edge type of degree distributions (DDs), we propose the probabilistic peeling decoding (P-PD) algorithm, which models the decoding step at every GC node as an instance of a Bernoulli random variable with a successful decoding probability that depends on both the GC block code and its decoding algorithm. The P-PD asymptotic performance over the BEC can be efficiently predicted using standard techniques for LDPC codes such as density evolution (DE) or the differential equation method. Furthermore, for a class of GLDPC ensembles, we demonstrate that the simulated P-PD performance accurately predicts the actual performance of the GLPDC code under ML decoding at GC nodes. We illustrate our analysis for GLDPC code ensembles with regular and irregular DDs. In all cases, we show that a large fraction of GC nodes is required to reduce the original gap to capacity, but the optimal fraction is strictly smaller than one. We then consider techniques to further reduce the gap to capacity by means of random puncturing, and the inclusion of a certain fraction of generalized variable nodes in the graph.
A generalized low-density parity check code (GLDPC) is a low-density parity check code in which the constraint nodes of the code graph are block codes, rather than single parity checks. In this paper, we study GLDPC codes which have BCH or Reed-Solomon codes as subcodes under bounded distance decoding (BDD). The performance of the proposed scheme is investigated in the limit case of an infinite length (cycle free) code used over a binary erasure channel (BEC) and the corresponding thresholds for iterative decoding are derived. The performance of the proposed scheme for finite code lengths over a BEC is investigated as well. Structures responsible for decoding failures are defined and a theoretical analysis over the ensemble of GLDPC codes which yields exact bit and block error rates of the ensemble average is derived. Unfortunately this study shows that GLDPC codes do not compare favorably with their LDPC counterpart over the BEC. Fortunately, it is also shown that under certain conditions, objects identified in the analysis of GLDPC codes over a BEC and the corresponding theoretical results remain useful to derive tight lower bounds on the performance of GLDPC codes over a binary symmetric channel (BSC). Simulation results show that the proposed method yields competitive performance with a good decoding complexity trade-off for the BSC.
Generalized low-density parity-check (GLDPC) codes, where the single parity-check (SPC) nodes are replaced by generalized constraint (GC) nodes, are known to offer a reduced gap to capacity when compared with conventional LDPC codes, while also maintaining linear growth of minimum distance. However, for certain classes of practical GLDPC codes, there remains a gap to capacity even when utilizing blockwise decoding algorithm at GC nodes. In this work, we propose to optimize the design of GLDPC codes where the GC nodes are decoded with a trellis-based bit-wise Bahl-Cocke-Jelinek- Raviv (BCJR) component decoding algorithm. We analyze the asymptotic threshold behavior of GLDPC codes and determine the optimal proportion of the GC nodes in the GLDPC Tanner graph.We show significant performance improvements compared to existing designs with the same order of decoding complexity.
In this study, the performance of generalized low-density parity-check (GLDPC) codes under the a posteriori probability (APP) decoder is analyzed. We explore the concentration, symmetry, and monotonicity properties of GLDPC codes under the APP decoder, extending the applicability of density evolution to GLDPC codes. On the binary memoryless symmetric channels, using the BEC and BI-AWGN channels as two examples, we demonstrate that with an appropriate proportion of generalized constraint (GC) nodes, GLDPC codes can reduce the original gap to capacity compared to their original LDPC counterparts. Additionally, on the BI-AWGN channel, we apply and improve the Gaussian approximation algorithm in the density evolution of GLDPC codes. By adopting Gaussian mixture distributions to approximate the message distributions from variable nodes and Gaussian distributions for those from constraint nodes, the precision of the channel parameter threshold can be significantly enhanced while maintaining a low computational complexity similar to that of Gaussian approximations. Furthermore, we identify a class of subcodes that can greatly simplify the performance analysis and practical decoding of GLDPC codes, which we refer to as message-invariant subcodes. Using the aforementioned techniques, our simulation experiments provide empirical evidence that GLDPC codes, when decoded with the APP decoder and equipped with the right fraction of GC nodes, can achieve substantial performance improvements compared to low-density parity-check (LDPC) codes.
Consider an ensemble of regular generalized LDPC (GLDPC) codes and assume that the same component code is associated with each parity check node. To decode a GLDPC code from the ensemble, we use the bit flipping bounded distance decoding algorithm, which is an extension of the bit flipping algorithm for LDPC codes. Previous work has shown conditions, under which, for a typical code in the ensemble with blocklength sufficiently large, a positive constant fraction of worst case errors can be corrected. In this work we first show that these requirements can be relaxed for ensembles with small left degrees. While previous work on GLDPC codes has considered expander graph arguments, our analysis formulates a necessary condition that the Tanner graph needs to satisfy for a failure event and then shows that the probability of this event vanishes for a sufficiently large blocklength. We then extend the analysis to random error correction and derive a lower bound on the fraction of random errors that can be corrected asymptotically. We discuss the extension of our results to non-binary GLDPC codes and present numerical examples.
The performance of generalized low-density parity-check (GLDPC) codes under binary and ternary message passing decoding (BMP/TMP) is analyzed from a density evolution (DE) perspective. At the check nodes, two types of local decoders are considered, namely optimum a-posteriori probability (APP) soft-input soft-output decoding, and bounded distance decoding (BDD). The purpose is to shed light on the performance loss incurred by BMP and TMP decoding of GLDPC codes with respect to unquantized belief propagation (BP) decoding. A DE analysis for irregular code ensembles is developed for all the algorithms, which allows obtaining the scaling coefficients needed for the variable node operation of BMP and TMP decoders. The stability analysis for the case of bounded distance decoding at the check nodes is derived. The asymptotic DE analysis is confirmed by the finite-length simulation results. For the codes analyzed in this paper, which rely on extended Hamming component codes, the study shows that under BMP decoding, BDD at the check nodes yields almost the same performance as optimum APP check node processing, while under TMP decoding the loss incurred by the sub-optimum BDD at check nodes is within 0.7 dB, when compared with APP decoding at the check nodes.