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GLDPC Codes Based on Polar Constraints and Their Near-Optimal Decoding

Binghui Shi, Yongpeng Wu, Yin Xu, Xiqi Gao, Xiaohu You, Wenjun Zhang

Published Dec 8, 2025• 1 citation
Publisher

Abstract

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.

DOI
10.1109/globecom59602.2025.11432491

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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.
Generalized low-density parity-check with polar component (GLDPC-PC) codes exhibit promising potential for 6G wireless systems with higher demands on reliability and data rate. The decoding method for GLDPC-PC codes involves belief propagation (BP), where the soft information of the component code is used as extrinsic information for check nodes. Although the Bahl-Cocke-Jelinek-Raviv (BCJR) decoding algorithm provides bitwise optimal soft information for linear block codes, its decoding complexity increases exponentially with the code length, making it impractical for applications. Therefore, the near-optimal soft-output successive cancellation list (SO-SCL) decoding with fixed scaling is proposed to reduce decoding complexity. However, the complexity of SO-SCL decoding still holds potential for further optimization. In this work, we propose an offline approximation of the unvisited probability codebook and a staged scaling strategy for the extrinsic information to reduce decoding complexity while maintaining low performance loss.
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.
Generalized low-density parity-check (GLDPC) codes have emerged as a promising class of error-correcting codes that extend the capabilities of traditional LDPC codes by incorporating more powerful component codes. This paper investigates an approach to construct long GLDPC codes using short component codes. In this scheme, the BCJR algorithm is used as a component decoder and integrated into the belief propagation algorithm. Simulation results demonstrate that constructions incorporating quasi-cyclic LDPC codes yield the best performance. Compared to the conventional decoding approach for LDPC codes, the proposed generalized constructions demonstrated better bit and frame error rate performance. This approach is promising for communication systems that require low latency. A comparison is also provided with LDPC codes constructed using the Progressive Edge-Growth (PEG) algorithm.
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.
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.
We describe a novel approach to interpret a polar code as a low-density parity-check (LDPC)-like code with an underlying sparse decoding graph. This sparse graph is based on the encoding factor graph of polar codes and is suitable for conventional belief propagation (BP) decoding. We discuss several pruning techniques based on the check node decoder (CND) and variable node decoder (VND) update equations, significantly reducing the size (i.e., decoding complexity) of the parity-check matrix. As a result, iterative polar decoding can then be conducted on a sparse graph, akin to the traditional well-established LDPC decoding, e.g., using a fully parallel sum-product algorithm (SPA). This facilitates the systematic analysis and design of polar codes using the well-established tools known from analyzing LDPC codes. We show that the proposed iterative polar decoder has a negligible performance loss for short-to-intermediate codelengths compared to Arikan's original BP decoder. Finally, the proposed decoder is shown to benefit from both reduced complexity and reduced memory requirements and, thus, is more suitable for hardware implementations.