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Non-Binary QC-LDPC Codes with Large Girth Devoid of Dominant Absorbing Sets

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

Published Oct 15, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.36227/techrxiv.176054693.32290404/v1

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International Conference on Instrumentation and Measurement, Computer, Communication and Control (IMCCC)•Jul 1, 2016•1 citation
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.
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.
This study proposes an explicit construction method for quantum quasi-cyclic low-density parity-check (QC-LDPC) codes with a girth of 12. The proposed method designs parity-check matrices that maximize the girth while maintaining an orthogonal structure suitable for quantum error correction. By utilizing algebraic techniques, short cycles are eliminated, which improves error correction performance. Additionally, this method is extended to non-binary LDPC codes and spatially-coupled LDPC codes, demonstrating that both the girth and orthogonality can be preserved. The results of this study enable the design of high-performance quantum error-correcting codes without the need for random search.
Multitype quasi-cyclic (QC) low-density parity-check (LDPC) codes are a class of protograph LDPC codes lifted cyclically from protographs with multiple edges, represented by two weight and slope matrices. For a given weight-matrix, an approach is proposed to find the maximum-achievable girthgmax⁡g_{\max }gmax​of the corresponding multitype QC-LDPC codes by some inevitable chains having less complexity than the existing methods. This advantage leads to some new patterns of the weight matrices such that the corresponding codes have some improvements in terms of the maximum-achievable girths or the minimum-distance upper-bounds. In continue, for a given weight-matrix with maximum-achievable girthgmax⁡g_{\max }gmax​, some slope-matrices are constructed by a depth-first search algorithm for which the corresponding multitype QC-LDPC codes with even girthggg,g≤gmax⁡g\le g_{\max }g≤gmax​, have smaller lengths, higher rates, or larger minimum-distances than the state-of-the-art achievements. Simulation results show that the constructed codes have some error-rate advantages than PEG, random-like, CCSDS, and 802.11n/ac IEEE standard LDPC codes.
There is an important breakthrough in the field of forward error correction techniques, low-density parity-check (LDPC) codes. These codes have potential to approach the Shannon limit with reliable performance on a given channel, known as the channel capacity. Recently, researchers pay much more attention towards LDPC codes because its performance is better than Turbo codes. However, the LDPC code is the best when it is an irregular code and uses with a very large block-length. Although, the short block-length LDPC codes perform not so well, they are easy to implement in various practical applications. Therefore, there is still a demand of new development on the encoding side of LDPC codes for various ranges of code rate and code length so as to capable of using in many applications.This thesis proposes the development of quasi-cyclic (QC) LDPC codes for column weight 3. In the first part of this thesis, we propose a new construction algorithm of QC-LDPC codes for medium to large block-length by combining QC-LDPC codes of small block-length as their component codes, via Chinese remainder theorem (CRT). Such component codes were constructed by permuting each column block sequentially to attain the desire local girth. After combining all component codes to generate an expanded parity-check matrix, the resulting girth is greater than or at least equal to the maximum girth of the component codes. Simulation results show that our proposed construction method of the parity-check matrix significantly outperforms the other well-known existing methods in terms of low error-floor, simple structure, high performance, and can reduce encoding complexity.In addition, this thesis also proposes two new construction methods for QC-LDPC codes, namely a base matrix method and a subtraction based method. A base matrix based method is a simple, less computational complexity method for constructing the exponent matrix (3, K) of girth 8, 10 and 12 of QC-LDPC codes. Another method is a subtraction method, which has a similar exponent matrix as a base matrix method and has a girth at least 8. Results indicate that the LDPC codes constructed from these two methods have flexibility for arbitrary block-column length K and have a similar BER performance if compared to existing methods.
Short cycles in the Tanner graph of a given LDPC code has negative effect on the performance of iterative decoding. To improve the performance, we presented an approach for constructing Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) codes whose associated Tanner graph has large girth and a small amount of short cycles. Each column block of the parity check matrix was designed by adding circulant permutation sub-matrices one after another, provided the current parity check matrix has minimal number of short cycles. The condition for the proposed QC-LDPC codes to have girth at least 6 was also analyzed. Regular and irregular QC-LDPC codes with various lengths and rates can be constructed. Simulations show that the proposed QC-LDPC codes have less short cycles and better performance than existing progressive edge growth based QC-LDPC codes.