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arXiv:2507.12073 · IEEE Transactions on Information Theory

On the Error Correction of Iterative Bounded Distance Decoding of Generalized LDPC Codes

David Burshtein

Published Sep 16, 2025• 0 citations
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Abstract

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.

DOI
10.1109/tit.2025.3609965

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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 iterative bit flipping algorithm is applied to the standard regular low-density parity-check (LDPC) code ensemble. In the past, it was shown, for a typical code in the ensemble with left degree at least five and block length sufficiently large, that this algorithm can correct a linear (in the block length) number of worst case errors. In this paper, this result is extended to the case where the left degree is at least four. For the case where the left degree is larger than four, an improvement, compared to existing results, of several orders of magnitude is obtained on the fraction of worst case errors that can be corrected. It is also shown how the results can be further improved when random errors produced by the channel (as opposed to worst case errors) are considered.
The relation between the girth and the guaranteed error correction capability of¿-left-regular low-density parity-check (LDPC) codes when decoded using the bit flipping (serial and parallel) algorithms is investigated. A lower bound on the size of variable node sets which expand by a factor of at least3 ¿/4is found based on the Moore bound. This bound, combined with the well known expander based arguments, leads to a lower bound on the guaranteed error correction capability. The decoding failures of the bit flipping algorithms are characterized using the notions of trapping sets and fixed sets. The relation between fixed sets and a class of graphs known as cage graphs is studied. Upper bounds on the guaranteed error correction capability are then established based on the order of cage graphs. The results are extended to left-regular and right-uniform generalized LDPC codes. It is shown that this class of generalized LDPC codes can correct a linear number of worst case errors (in the code length) under the parallel bit flipping algorithm when the underlying Tanner graph is a good expander. A lower bound on the size of variable node sets which have the required expansion is established.
We apply the iterative bit flipping algorithm to the standard regular low-density parity-check (LDPC) code ensemble. In the past it was shown, for a typical code in the ensemble with left degree at least five and block length sufficiently large, that this algorithm can correct all error patterns with some linear (in the block length) number of errors. We extend this result to the case where the left degree is at least four. For the case where the left degree is larger than four, we obtain an improvement of several orders of magnitude compared to the existing results on the fraction of worst case errors that can be corrected. We also show how our results can be improved when we consider random errors (as opposed to worst case errors) produced by the channel.
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.
The relation between the girth and the guaranteed error correction capability of γ-left regular LDPC codes when decoded using the bit flipping (serial and parallel) algorithms is investigated. A lower bound on the size of variable node sets which expand by a factor of at least 3γ/4 is found based on the Moore bound. An upper bound on the guaranteed error correction capability is established by studying the sizes of smallest possible trapping sets. The results are extended to generalized LDPC codes. It is shown that generalized LDPC codes can correct a linear fraction of errors under the parallel bit flipping algorithm when the underlying Tanner graph is a good expander. It is also shown that the bound cannot be improved when γ is even by studying a class of trapping sets. A lower bound on the size of variable node sets which have the required expansion is established.
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.