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Array-Based Spatially Coupled LDPC Codes Using Generalized Behrend Sequences

Suresh Kumar Venkataramanappa, Pavan Kumar, Deepak Mishra, Shayan Srinivasa Garani

Published Oct 30, 2025• 0 citations
Publisher

Abstract

We present a novel construction of quasi-cyclic (QC) low-density parity-check (LDPC) codes with arbitrary column weight and girth at least eight. Furthermore, we demonstrate that in the case of column weight three, the absorbing sets (3, 3) and (4, 2) can be entirely avoided in our construction, resulting in improved code performance. Building on the strong performance of our constructed QC-LDPC codes, we extend them to design several families of spatially coupled (SC) QC-LDPC codes, which further enhance decoding performance under windowed decoding. Simulation results under the additive white Gaussian noise (AWGN) channel demonstrate that the proposed QC-LDPC codes outperform the LDPC codes part of the space data systems standards. Additionally, we benchmark our proposed QC-LDPC and SC-QC-LDPC codes against existing girth-8 codes, observing significant improvements in the decoding performance of SC-QC-LDPC codes up to 1.1 dB compared to the protographbased construction by Shiyun Mo et al. [IEEE Trans. Commun.], while maintaining suitability for hardware implementation. The findings of this work contribute to the ongoing development of efficient LDPC codes suitable for ultra-reliable low-latency communications and area-efficient architecture designs.

DOI
10.36227/techrxiv.175382813.34829210/v2

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We present a novel construction of quasi-cyclic (QC) low-density parity-check (LDPC) codes with arbitrary column weight and girth at least eight. Furthermore, we demonstrate that in the case of column weight three, the absorbing sets (3, 3) and (4, 2) can be entirely avoided in our construction, resulting in improved code performance. Building on the strong performance of our constructed QC-LDPC codes, we extend them to design several families of spatially coupled (SC) QC-LDPC codes, which further enhance decoding performance under windowed decoding. Simulation results under the additive white Gaussian noise (AWGN) channel demonstrate that the proposed QC-LDPC codes outperform the LDPC codes part of the space data systems standards. Additionally, we benchmark our proposed QC-LDPC and SC-QC-LDPC codes against existing girth-8 codes, observing significant improvements in the decoding performance of SC-QC-LDPC codes up to 1.1 dB compared to the protographbased construction by Shiyun Mo et al. [IEEE Trans. Commun.], while maintaining suitability for hardware implementation. The findings of this work contribute to the ongoing development of efficient LDPC codes suitable for ultra-reliable low-latency communications and area-efficient architecture designs.
We present a novel construction of quasi-cyclic (QC) low-density parity-check (LDPC) codes with arbitrary column weight and girth at least eight. Furthermore, we demonstrate that in the case of column weight three, the absorbing sets (3, 3) and (4, 2) can be entirely avoided in our construction, resulting in improved code performance. Building on the strong performance of our constructed QC-LDPC codes, we extend them to design several families of spatially coupled (SC) QC-LDPC codes, which further enhance decoding performance under windowed decoding. Simulation results under the additive white Gaussian noise (AWGN) channel demonstrate that the proposed QC-LDPC codes outperform the LDPC codes part of the space data systems standards. Additionally, we benchmark our proposed QC-LDPC and SC-QC-LDPC codes against existing girth-8 codes, observing significant improvements in the decoding performance of SC-QC-LDPC codes up to 1.1 dB compared to the protographbased construction by Shiyun Mo et al. [IEEE Trans. Commun.], while maintaining suitability for hardware implementation. The findings of this work contribute to the ongoing development of efficient LDPC codes suitable for ultra-reliable low-latency communications and area-efficient architecture designs.
We present a novel construction of quasi-cyclic (QC) low-density parity-check (LDPC) codes with arbitrary column weight and girth at least eight. By carefully designing the parity-check matrix, our approach ensures both structural flexibility and improved error-correction capabilities. We evaluate the performance of these codes under the additive white Gaussian noise (AWGN) channel. Simulation results demonstrate that the proposed QC-LDPC codes outperform the protograph LDPC codes included in the Consultative Committee for Space Data Systems (CCSDS) telecommand standards. Additionally, we benchmark our proposed codes against existing short-length codes with girth 8, observing improvements in the decoding performance while ensuring suitability for hardware implementation. The findings of this work contribute to the ongoing development of efficient LDPC codes suitable for ultra-reliable low-latency communications and area-efficient architecture designs for the Internet of Things (IoT), machine-to-machine (M2M), and satellite communications.
Non-binary (NB) low-density parity-check (LDPC) codes offer superior error-correcting performance compared to binary LDPC codes, especially for short and medium lengths. Constructing such codes with large girth remains challenging due to various combinatorial constraints. We present a novel algebraic and graph-theoretic construction of non-binary quasi-cyclic (QC) LDPC codes with column weight two and girth 12, which is known to be largest achievable girth for array-based QC-LDPC codes. In constructing girth-12 (2, dc) NB-QC-LDPC codes, our method attains the minimum possible code length for dc < 10. Motivated by the fact that non-binary spatially coupled (SC) QC-LDPC codes achieve optimal performance at column weight three, we further construct NB-SC-QC-LDPC codes with girth ≥ 8 while effectively reducing the number of 8-cycles. Specifically, for (3, dc) codes, our design eliminates dominant absorbing sets such as the (3, 3) and (4, 2), eliminates the (5, 1) elementary absorbing sets (EAS), and eliminates almost all (6, 2) EAS, thereby directly enhancing the error-correction performance. The proposed NB-QC-LDPC codes achieve performance comparable to NB-LDPC codes used in the BeiDou satellite navigation system, while requiring a significantly smaller parity-check matrix storage footprint, for instance, around a 44× reduction in storage requirements with respect to BeiDou rate-1/2 (88, 44) code. Also, the proposed rate-1/2 code with length n = 684 provide a 0.4 dB gain over non-binary near-regular QC-LDPC code with same parameters proposed by Huang et al. [IEEE Trans. Commun.] while requiring a smaller parity-check matrix storage footprint. Furthermore, the proposed rate-1/2 NB-SC-QC-LDPC code having length n = 4608 achieve 0.1 dB coding gain compared to existing protograph constructions while requiring a significantly smaller parity-check matrix storage footprint, i.e., around a 192× reduction.
Spatially coupled (SC) low-density parity-check (LDPC) codes can achieve capacity approaching performance with low message recovery latency when using sliding window (SW) decoding. An SC-LDPC code constructed from a protograph can be generated by first coupling a chain of block protographs and then lifting the coupled protograph using permutation matrices. In this paper, we introduce a systematic design to eliminate 4-cycles in a coupled protograph. Further using a quasi-cyclic (QC) lifting, we introduce a procedure for constructing QC-SC-LDPC codes of girth at least eight. This can be interpreted as a multi-stage graph lifting process that yields a greater flexibility in designing QC-SC-LDPC codes with a large girth than previous approaches. Simulation results show the design leads to improved decoding performance, particularly in the error floor, compared to random constructions. Finally, we determine the minimum coupling width required to eliminate 4-cycles in a coupled protograph.
Spatially coupled (SC) low-density parity-check (LDPC) codes can achieve capacity approaching performance with low message recovery latency when using sliding window (SW) decoding. An SC-LDPC code constructed from a protograph can be generated by first coupling a chain of block protographs and then lifting the coupled protograph using permutation matrices. This paper introduces a systematic design of SC-LDPC codes to eliminate 4-cycles in the coupled photograph. Using a quasi-cyclic (QC) lifting, we obtain QC-SC-LDPC codes of girth at least eight. Coupling a chain of block protographs implies spreading edges from one protograph to the others. Our protograph-based design can be viewed as guiding the edge spreading and also the graph-lifting process. Simulation results show the design leads to improved decoding performance, particularly in the error floor, compared to random designs.
Quasi-cyclic low-density parity-check (QC-LDPC) codes are known to have algebraic structure, which suits hardware implementation. However, low girth makes their performances worse than random structured LDPC codes. Based on independent row-column mapping sequence, this work proposed a generalized method of constructing QC-LDPC codes with girth eight, which enlarges the class of girth eight QC-LDPC codes to a fairly general series. Simulation results show that the bit error rate of the constructed codes is close to that of progressive edge growth (PEG) based QC-LDPC codes under iterative decoding.