一种基于有限域的二进制和多进制空间耦合LDPC码构造方法
An Approach for Constructing Binary and Nonbinary Spatially-Coupled LDPC Codes Based on Finite Fields
摘要: 空间耦合LDPC (Spatially-Coupled LDPC, SC-LDPC)码由于阈值饱和特性，被证明是未来无线通信系统的有力候选码型。SC-LDPC码是一种卷积LDPC码，在二元无记忆对称信道下采用置信传播算法时具有逼近香农限的性能。本文提出了一种构造二进制和多进制空间耦合LDPC码的统一方法。基于有限域GF(q)上的本原元、加法子群、乘法子群，构造了6类二进制和多进制空间耦合准循环LDPC码(Spatially-Coupled Quasi­Cyclic LDPC, SC-QC-LDPC)，用此构造的LDPC码的围长至少为6。
Abstract: Spatially-coupled LDPC codes are demonstrated to strong candidates for future optical communications systems due to the threshold saturating property. Spatially-coupled LDPC code is a kind of convolutional LDPC code. It has the performance of approaching Shannon limit when using belief propagation decoding algorithm in binary memoryless symmetric channel. A unified approach for constructing binary and nonbinary spatially-coupled LDPC codes is presented in this paper. Six classes of binary and nonbinary spatially-coupled Quasi-Cyclic LDPC codes are constructed based on primitive elements, additive subgroups, multiplicative subgroups of finite fields. Moreover, the girth of these codes is greater than or equal to 6.
文章引用：梁宇, 杨卫华, 李玉瑛. 一种基于有限域的二进制和多进制空间耦合LDPC码构造方法[J]. 应用数学进展, 2021, 10(2): 575-586. https://doi.org/10.12677/AAM.2021.102063

参考文献

[1] Felstrom, A.J. and Zigangirov, K. (1999) Time-Varying Periodic Convolutional Codes with Low-Density Parity-Check Matrix. IEEE Transactions on Information Theory, 45, 2181-2191. [Google Scholar] [CrossRef] 
[2] Gallager, R.G. (1962) Low-Density Parity-Check Codes. IRE Transactions on Information Theory, 8, 21-28. [Google Scholar] [CrossRef] 
[3] Kudekar, S., Richardson, T. and Urbanke, R. (2011) Threshold Saturation via Spatial Coupling: Why Convolutional LDPC Ensembles Perform So Well over the BEC. IEEE Transactions on Information Theory, 57, 803-834. [Google Scholar] [CrossRef] 
[4] Kudekar, S., Measson, C., Richardson, T. and Urbankez, R. (2010) Threshold Saturation on BMS Channels via Spatial Coupling. 6th International Symposium on Turbo Codes and Iterative Information Processing, Brest, 6-10 September 2010, 309-313. [Google Scholar] [CrossRef] 
[5] Yedla, A., Jian, Y.-Y., Nguyen, P. and Pfifister, H. (2012) A Simple Proof of Threshold Saturation for Coupled Scalar Recursions. 2012 7th International Symposium on Turbo Codes and Iterative Information Processing, Gothenburg, 27-31 August 2012, 51-55. [Google Scholar] [CrossRef] 
[6] Lentmaier, M., Sridharan, A., Costello Jr., D.J. and Zigangirov, K. (2010) Iterative Decoding Threshold for LDPC Convolutional Codes. IEEE Transactions on Information Theory, 56, 5274-5289. [Google Scholar] [CrossRef] 
[7] Huang, K., Mitchell, D.G.M., Wei, L., Ma, X. and Costello, D. (2015) Performance Comparison of LDPC Block and Spatially Coupled Codes over GF(q). IEEE Transactions on Communications, 63, 592-604. [Google Scholar] [CrossRef] 
[8] Iyengar, A., Papaleo, M., Siegel, P., Wolf, J., Vanelli-Coralli, A. and Corazza, G. (2012) Windowed Decoding of Protograph-Based LDPC Convolutional Codes over Erasure Channels. IEEE Transactions on Information Theory, 58, 2303-2320. [Google Scholar] [CrossRef] 
[9] Andriyanova, I. and Graelli Amat, A. (2013) Threshold Saturation for Nonbinary SC-LDPC Codes on the Binary Erasure Channel. IEEE Transactions on Information Theory, 62, 2622-2638. [Google Scholar] [CrossRef] 
[10] Pusane, A., Smarandache, R., Vontobel, P. and Costello, D. (2011) Deriving Good LDPC Convolutional Codes from LDPC Block Codes. IEEE Transactions on Information Theory, 57, 835-857. [Google Scholar] [CrossRef] 
[11] Fossorier, M. (2004) Quasi Cyclic Low-Density Parity-Check Codes from Circulant Permutation Matrices. IEEE Transactions on Information Theory, 50, 1788-1793. [Google Scholar] [CrossRef] 
[12] Li, Z., Chen, L., Zeng, L., Lin, S. and Fong, W. (2005) Efficient Encoding of Quasi-Cyclic Low-Density Parity-Check Codes. IEEE Transactions on Communications, 53, 1973-1973. [Google Scholar] [CrossRef] 
[13] Chen, Y. and Parhi, K. (2004) Overlapped Message Passing for Quasi-Cyclic Low-Density Parity Check Codes. IEEE Transactions on Circuits and Systems I: Regular Papers, 51, 1106-1113. [Google Scholar] [CrossRef] 
[14] Uchikawa, H. Kasai, K. and Sakaniwa, K. (2011) Design and Performance of Rate-Compatible Non-Binary LDPC Convolutional Codes. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, E94, 2135-2143.
[15] Ryan, W. and Lin, S. (2009) Channel Codes: Classical and Modern. Cambridge University Press, Cambridge. [Google Scholar] [CrossRef] 
[16] Lan, L., Zeng, L., Tai, Y., Chen, L., Lin, S. and Abdel-Ghaffar, K. (2007) Construction of Quasi-Cyclic LDPC Codes for AWGN and Binary Erasure Channels: A Finite Field Approach. IEEE Transactions on Information Theory, 53, 2429-2458. [Google Scholar] [CrossRef] 
[17] Zhang, L., Huang, Q., Lin, S., Abdel-Ghaffar, K. and Blake, I.F. (2010) Quasi-Cyclic LDPC Codes: An Algebraic Construction, Rank Analysis, and Codes on Latin Squares. IEEE Transactions on Communications, 58, 3126-3139. [Google Scholar] [CrossRef] 
[18] Dolecek, L., Divsalar, D., Sun, Y. and Amiri, B. (2014) Non-Binary Protograph-Based LDPC Codes: Enumerators, Analysis, and Designs, Information Theory. IEEE Transactions on Information Theory, 60, 3913-3941. [Google Scholar] [CrossRef] 
[19] Chandrasetty, V.A., Johnson, S.J. and Lechner, G. (2013) Memory Efficient Decoders Using Spatially Coupled Quasi-Cyclic LDPC Codes. Clinical Orthopaedics and Related Research, arXiv:1305.5625
[20] Hu, X.-Y., Eleftheriou, E. and Arnold, D.-M. (2005) Regular and Irregular Progressive Edge-Growth Tanner Graphs. IEEE Transactions on Information Theory, 51, 386-398. [Google Scholar] [CrossRef] 
[21] Li, J., Liu, K., Lin, S., Abdel-Ghaffar, K. and Ryan, W.E. (2015) An Unnoticed Strong Connection between Algebraic-Based and Protograph-Based LDPC Codes. 2015 Information Theory and Applications Workshop, San Diego, 1-6 February 2015, 36-45. [Google Scholar] [CrossRef] 
[22] Wei, L., Koike-Akino, T., Mitchell, D., Fuja, T. and Costello, D.J. (2014) Threshold Analysis of Non-Binary Spatially-Coupled LDPC Codes with Windowed Decoding, in Information Theory (ISIT). 2014 IEEE International Symposium on, Honolulu, 29 June-4 July 2014, 881-885. [Google Scholar] [CrossRef] 

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