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arXiv:2505.08030 · IEEE Wireless Communications Letters

Generalized LDPC Codes With Low-Complexity Decoding and Fast Convergence

Dawit Simegn, Dmitry Artemasov, Kirill Andreev, Pavel S. Rybin, Alexey Frolov

Published Aug 20, 2025• 0 citations
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Abstract

We consider generalized low-density parity-check (GLDPC) codes with component codes that are duals of Cordaro–Wagner codes. Two efficient decoding algorithms are proposed: one based on Hartmann–Rudolph processing, analogous to Sum-Product decoding, and another based on evaluating two hypotheses per bit, referred to as the Min-Sum decoder. Both algorithms are derived using latent variables and a appropriate message-passing schedule. A quantized, protograph-based density evolution procedure is used to optimize GLDPC codes for Min-Sum decoding. Compared to 5G LDPC codes, the proposed GLDPC codes offer similar performance at 50 iterations and significantly better convergence and performance at 10 iterations.

DOI
10.1109/lwc.2025.3600912

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We consider generalized low-density parity-check (GLDPC) codes with component codes that are duals of Cordaro–Wagner codes. Two efficient decoding algorithms are proposed: one based on Hartmann–Rudolph processing, analogous to Sum-Product decoding, and another based on evaluating two hypotheses per bit, referred to as the Min-Sum decoder. Both algorithms are derived using latent variables and a appropriate message-passing schedule. A quantized, protograph-based density evolution procedure is used to optimize GLDPC codes for Min-Sum decoding. Compared to 5G LDPC codes, the proposed GLDPC codes offer similar performance at 50 iterations and significantly better convergence and performance at 10 iterations.
In this letter, we investigate the min-sum (MS) and neural MS (NMS) decoding algorithms for generalized low-density parity-check (GLDPC) codes. Although the MS decoder is much simpler than the a posteriori probability (APP) decoder commonly used for GLDPC codes, the MS decoder has not been considered mainly due to its inferior decoding performance. However, we show that the performance can be improved by i) employing the NMS decoding algorithm and ii) optimizing the component parity check matrix (PCM). For the four representative short GLDPC codes in the literature, experimental results show that the NMS decoding performance with the optimized component PCM significantly outperforms the MS decoding performance and even outperforms the APP decoding performance for some cases.
In this work, we introduce the integration of generalized low-density parity-check (GLDPC) codes with short polar component codes, termed GLDPC codes with polar component codes (GLDPC-PC). A recently proposed soft-input soft-output (SISO) decoder for polar-like codes enables effective iterative belief propagation decoding for GLDPC-PC. This SISO decoder after a post-processing exhibits little performance loss to the optimal SISO decoder when all the variable nodes have relatively low degrees. A three-step method is introduced to design protograph-based GLDPC codes. The constructed GLDPC codes are compared with 5G LDPC codes. They exhibit little performance loss in the waterfall region and possess better error floor with less iterations.
Abstract: In this paper, a fast-convergence algorithm is proposed for the decoding of low-density parity-check (LDPC) codes. By putting more weight on the message update from check node to variable node in current iteration, the proposed method speeds up the decoding significantly. Simulation results show that compared with the standard layered min-sum decoding algorithm, the proposed scheme may reduce about 1/6 iteration numbers for LDPC codes used in DVB-S2, with hardware complexity overhead less than 2%. Key–Words: LDPC code, fast convergence decoding algorithm, layered decoding, min-sum algorithm, hardware implementation 1
Due to the more complex constraints, generalized low-density parity-check (GLDPC) codes can achieve better error performance but require much higher decoding complexities compared with the standard LDPC codes. In this paper, single-parity-check product-codes (SPC-PCs) are considered as the constituent codes in the super check nodes of a GLDPC code. Moreover, turbo iterations are used in decoding the SPC-PCs. The error performance and the decoder complexity of the proposed GLDPC code are compared with other LDPC code and GLDPC code.
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.
Low-density parity-check (LDPC) codes have received significant attention due to their near-Shannon-limit error performance. The belief propagation (BP) algorithm is the best-known decoding algorithm for LDPC codes. However it often needs several tens or hundreds of iterations to converge. Although fast convergence BP algorithms have been proposed, there is neither an easy tool to analyze their performance nor a standard VLSI architecture to implement them. Moreover, there exists a trade-off between waterfall (i.e., threshold) and error floor (i.e., minimum pseudo distance) performances for LDPC codes. For generalized LDPC (GLDPC) codes, better minimum distance properties can be achieved but at the expense of a rate loss. This research investigates performance analysis and hardware implementation of LDPC codes using fast convergence algorithms. Closed-form extrinsic mutual information transfer (EXIT) functions for several fast convergence BP decoding algorithms are proposed. It is shown theoretically that these algorithms converge faster than standard BP, while the corresponding thresholds remain unchanged. Since these algorithms are of serial nature, their parallelization without compromising convergence speed and error performance is also investigated. This result is useful in hardware implementation because in practice partly parallel decoding schemes are often used. This result also sheds some light on how to group nodes in order to preserve performance. The VLSI architecture of a particular decoding algorithm for LDPC codes is proposed. This approach can be adopted in many standards to achieve a large throughput. Doubly GLDPC (DGLDPC) codes are proposed to compensate the rate loss of GLDPC codes. EXIT charts are used to analyze DGLDPC codes and EXIT functions for variable component codes are thoroughly discussed. Based on EXIT charts, differential evolution is used to optimize their threshold. Ensemble weight enumerators are generalized to protograph-based DGLDPC codes. Simulation results show that DGLDPC codes constructed based on these results can improve the performance of their LDPC and GLDPC counterparts in both the waterfall region and the error floor region.