基于动态消息调度的LDPC码及数字喷泉码改进译码算法研究
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摘要
随着无线通信的飞速发展,高速率的数据传输以及多媒体业务的急速增多,在提高信息传输效率的基础上,如何保证通信过程的可靠性显得尤为重要,这对纠错编码提出了更高的要求。低密度奇偶校验码(LDPC)具有逼近香农极限的纠错性能,且有描述方便、易理论分析和译码简单的特点,适合硬件实现；数字喷泉码作为一种无固定码率码,可以根据信道的实时状况,自适应地匹配编码方案。置信传播(BP)译码是LDPC码和数字喷泉码两种编码的基本译码算法,但传统的BP译码算法,不能充分利用信息传递过程中的更新信息。基于置信传播的动态译码算法,为实现每次信息传递提供更多的置信度提供了一个有效的途径,近来吸引了大量的学者对其进一步的研究。
论文首先简要介绍了LDPC码和LT码的基本概念,包括码字的表示、特性以及编码过程,接着详细介绍了两种码字通用的迭代译码算法,即BP译码算法的原理和译码流程。在此基础上,论文分析和讨论了影响BP译码算法的关键因素,分析了消息调度对BP译码算法的影响。接着,论文详细分析了基于残差的置信传播动态的消息调度算法,并结合Tanner图详尽分析了基于残差的RBP、 NW-RBP和VC-RBP三种动态消息调度算法的具体调度过程,讨论了每种调度算法的特点,阐释了相关消息调度克服陷阱集的过程,采用仿真研究的方法,分析并讨论了不同的动态消息调度策略对LDPC码及LT码译码性能的影响。
论文针对LDPC码和LT码的相关仿真研究分析表明,动态消息调度对基于置信传播的迭代译码算法的改进高度依赖于编码构造。一般来说,对于中、短码长的码字,如果码字的Tanner图中变量节点和校验节点的度数分布相对均匀,且无大量度为1的节点以及过多的四环时,采用基于动态消息调度的BP译码算法可以显著提高译码性能；对于度分布极度不均匀或四环过多的中、短码长码字,或者度分布为1的节点过多时,采用基于动态消息调度对BP译码算法的改进并不明显。此外,论文的分析结果说明,在实际系统应用时,应结合编码结构,合理选择相关消息调度机制以充分发挥基于动态消息调度的改进译码算法所能带来的潜在的性能改进。
With the rapid development of wireless communication, high-speed data transmission and multimedia sevices increase rapidly. Therefore, how to ensure the reliability of communication process is particularly important on the basis of information transmitted, and need higher demand on the error-correcting codes. Low-density parity-check (LDPC)codes can approach the Shannon limit error correction performance, which is described conveniently and easy theoretical analysis as well as decoding. Meanwhile, it is quite Suitable for hardware implementation. On the other hand, digital fountain code as an efficient rateless codes, it can adaptive matching encoding scheme, according to channel status from time to time. Belief Propagation(BP) algorithm is decoding basis of these two codes. However, conventional BP can't pass the latest updated message to adjacent edges. Then based on the informed dynamic scheduling(IDS) for BP decoding, there is more confidence message for every message passing.
In this paper, first briefly introduce the basic concepts of LDPC codes and LT codes, including representations, feature and encoding processing. Then detailed iterative decoding algorithm is described, that is principle and process of BP algorithm. On this basis, analyze two key factors affecting the BP decoding algorithm, as well as informed dynamic schedules (IDS). Next, discuss and analyze the specific process of BP decoding algorithm based on residual value of IDS. On this basis, combined to Tanner gragh, mainly introduce three algorithm, including Residual Belief Propagation(RBP) and Node-wise Residual Belief Propagation (NW-RBP), which based on the differences between the value of the check nodes to variable nodes message before and after an update, variable nodes to check nodes Residual Belief Propagation(VCRBP), which based on the he differences between the value of the variable nodes to check nodes message before and after an update. Moreover, discuss the characteristics of each scheduling algorithm, and expound process that the relevant IDS strategies overcome "trapping sets". By simulation study, analyze and compare the decoding performance on different IDS strategies affecting on LDPC codes and LT codes.
Simulation study analysis, according to LDPC and LT codes, show that the performance for the BP decoding algorithm on IDS can be improved. However, it depend on coding constructs. Generally speaking, if a codeword is medium and short, and degree distribution relatively uniform in the Tanner graph, and there is no large number of variable nodes with one degree as well as girth, then the BP decoding on IDS can significantly improve decoding performance. Otherwise, if a codeword is medium and short, the performance improvement is not significant. So the papers of analysis results indicate that, in practice, combined with the coding structure, be reasonable to choose improved BP decoding algorithm on IDS strategies, in order to give full play to the role of IDS that can bring potential performance improvements.
论文首先简要介绍了LDPC码和LT码的基本概念,包括码字的表示、特性以及编码过程,接着详细介绍了两种码字通用的迭代译码算法,即BP译码算法的原理和译码流程。在此基础上,论文分析和讨论了影响BP译码算法的关键因素,分析了消息调度对BP译码算法的影响。接着,论文详细分析了基于残差的置信传播动态的消息调度算法,并结合Tanner图详尽分析了基于残差的RBP、 NW-RBP和VC-RBP三种动态消息调度算法的具体调度过程,讨论了每种调度算法的特点,阐释了相关消息调度克服陷阱集的过程,采用仿真研究的方法,分析并讨论了不同的动态消息调度策略对LDPC码及LT码译码性能的影响。
论文针对LDPC码和LT码的相关仿真研究分析表明,动态消息调度对基于置信传播的迭代译码算法的改进高度依赖于编码构造。一般来说,对于中、短码长的码字,如果码字的Tanner图中变量节点和校验节点的度数分布相对均匀,且无大量度为1的节点以及过多的四环时,采用基于动态消息调度的BP译码算法可以显著提高译码性能；对于度分布极度不均匀或四环过多的中、短码长码字,或者度分布为1的节点过多时,采用基于动态消息调度对BP译码算法的改进并不明显。此外,论文的分析结果说明,在实际系统应用时,应结合编码结构,合理选择相关消息调度机制以充分发挥基于动态消息调度的改进译码算法所能带来的潜在的性能改进。
With the rapid development of wireless communication, high-speed data transmission and multimedia sevices increase rapidly. Therefore, how to ensure the reliability of communication process is particularly important on the basis of information transmitted, and need higher demand on the error-correcting codes. Low-density parity-check (LDPC)codes can approach the Shannon limit error correction performance, which is described conveniently and easy theoretical analysis as well as decoding. Meanwhile, it is quite Suitable for hardware implementation. On the other hand, digital fountain code as an efficient rateless codes, it can adaptive matching encoding scheme, according to channel status from time to time. Belief Propagation(BP) algorithm is decoding basis of these two codes. However, conventional BP can't pass the latest updated message to adjacent edges. Then based on the informed dynamic scheduling(IDS) for BP decoding, there is more confidence message for every message passing.
In this paper, first briefly introduce the basic concepts of LDPC codes and LT codes, including representations, feature and encoding processing. Then detailed iterative decoding algorithm is described, that is principle and process of BP algorithm. On this basis, analyze two key factors affecting the BP decoding algorithm, as well as informed dynamic schedules (IDS). Next, discuss and analyze the specific process of BP decoding algorithm based on residual value of IDS. On this basis, combined to Tanner gragh, mainly introduce three algorithm, including Residual Belief Propagation(RBP) and Node-wise Residual Belief Propagation (NW-RBP), which based on the differences between the value of the check nodes to variable nodes message before and after an update, variable nodes to check nodes Residual Belief Propagation(VCRBP), which based on the he differences between the value of the variable nodes to check nodes message before and after an update. Moreover, discuss the characteristics of each scheduling algorithm, and expound process that the relevant IDS strategies overcome "trapping sets". By simulation study, analyze and compare the decoding performance on different IDS strategies affecting on LDPC codes and LT codes.
Simulation study analysis, according to LDPC and LT codes, show that the performance for the BP decoding algorithm on IDS can be improved. However, it depend on coding constructs. Generally speaking, if a codeword is medium and short, and degree distribution relatively uniform in the Tanner graph, and there is no large number of variable nodes with one degree as well as girth, then the BP decoding on IDS can significantly improve decoding performance. Otherwise, if a codeword is medium and short, the performance improvement is not significant. So the papers of analysis results indicate that, in practice, combined with the coding structure, be reasonable to choose improved BP decoding algorithm on IDS strategies, in order to give full play to the role of IDS that can bring potential performance improvements.
引文
[1]张霄竞,无线中继在下一代移动通信中的应用[J],电信科学,2007年第9期,PP.62-66.
[2]Proakis J.G, "Digital Communication",3rd edition, New Nork:Me Graw-Hill,1995.
[3]Stephen G Wilson. Digital Modulation and Coding. Prentice-Hall, 1996.
[4]T J Richardson and R L Urbanke. The Capacity of Low-density Parity-check Codes Under Message-passing Decoding. IEEE Trans. Information Theory, Vol.47, pp.599-618, Feb.2001
[5]R G Gallgager. Information Theory and Reliable Communication. Wiley New York, 1968.
[6]Xingkai Bao, Jing Li. Adaptive Network Coded Cooperation(ANCC) for Wireless Relay Networks: Matching Code-on-Graph with Network-on-Graph. IEEE Transactions on wireless communications, VOL.7, NO.2, February 2008.
[7]E K Hall, Stephen G Wilson. Design and Analysis of Turbo Codes on Rayleigh Fading Channel. IEEE Journal on Selected Areas in Communications, Vol.16, No.2, pp.160-174, Feb.1998.
[8]Hamming R. Error Detecting and Error Correcting Codes[J]. Bell System Technical Journal,1950,29(2):147-160.
[9]Viterbi A. Error Bounds for Convolutional Codes and An Asymptotically Optimum Decoding Algorithm [J]. IEEE Trans. Information on Theory, 1967,13(2):260-269.
[10]Goppa V. Algebraico-Geometric Codes [J]. Izvestiya:Mathematics,1983,21(1):75-91.
[11]Goppa V. Geometry and Codes [J]. D Reidel Pub Co, 1988.
[12]Berrou Claude. The Ten-Year-Old Turbo Codes are Entering into Service. IEEE Communications Magazine, 2003, (8):110-115.
[13]D J C MacKay and R N Neal. Near Shannon limit performance of low density parity check codes [J]. Electronic Letters, Aug.1996,32:1645 - 1646.
[14]R. Gallgager. Low-density parity-check codes. IRE Trans. Information Theory, pp.21-28, Jan.1962.
[15]R. M. Tanner. A recursive approach to low complexity codes. IEEE Trans. Information Theory, pp.533-547, Sept. 1981.
[16]D. MacKay and R. Neal. Good codes based on very sparse matrices. The Proceeding of the 5th IMA Conf. in Cryptography and Coding,, C. Boyd, Ed., Lecture Notes in Computer Science, pp.100-111, Berlin, Germany: Springer, 1995.
[17]D. MacKay. Good error correcting codes based on very sparse matrices. IEEE Trans. Information Theory, pp.399-431, March 1999.
[18]N. Alon and M. Luby. A linear time erasure-resilient code with nearly optimal recovery. IEEE Trans. Information Theory, pp.1732-1736, Nov.1996.
[19]S. Y. Chung, G. D. Forney, Jr., T. J. Richardson, and R. Urbanke. On the design of low-density parity check codes within 0.0045dB of the Shannon limit. IEEE Commun.. Letters,2001,5(2):58-60.
[20]Spielman D. A.. Finding Good LDPC Codes, http://spielman@math.mit.edu.
[21]IEEE Std 802.16e. Air interface for fixed and mobile broadband wireless access systems. IEEE P802.16e/D12 Draft. Oct 2005. pp.626-630.
[22]Luby, M.. LT codes. Foundations of Computer Science,2002. Proceedings, of The 43rd Annual IEEE Symposium on Foundations of Computer Science, pp.271-280, 2002.
[23]Shokrollahi, A..Raptor codes. IEEE Trans. Inform. Theory, vol.52, no.6, pp.2551-2567, June 2006.
[24]R.Palanki and J. S. Yedidia, "Rateless codes on Noisy Channels," Proc. of IEEE International Symposium on Information Theory, Jun.2004.
[25]Castura, J.; Mao, Y.. Rateless coding over fading channels. IEEE Communications Letters,, vol.10, no.1, pp.46-48, Jan 2006.
[26]O. Etesami and A. Shokrollahi. Raptor codes on binary memoryless symmetric channels. IEEE Trans. Inform. Theory, vol.52, no.5, pp2033-2051, May 2006.
[27]Robert J McEliece. The Theory of Information and Coding. Addison-Wesley, Reading, 1977.
[28]William E Ryan. An Introduction to LDPC Codes, Department of Electrical and Computer Engineering, The University of Arizona, Aug 2003
[29]文红,符初生,周亮LDPC码原理与应用电子科技大学出版社,2006年：1-44,55-63,97-112.
[30]D.J.C.Mackay. Good error-correcting codes based on very sparsed matrices, IEEETrans.Inform.Theory, Marl 999:399-433.
[31]D.Divsalar, H. Jin, and R. McEliece, Proc.36th Annual Allerton Conf. on Comm., Control, and Cmputingce. Coding theorems for turbo-like codes, pp.201-210, Sept.1998.
[32]Omid Etesami and Amin Shokrollahi, "Raptor codes on Binary Memoryless Symmetric Channels,", IEEE Trans. Inform. Theory, vol.52, no.5, pp.2033-2051, May.2006.
[33]E. Yeo, P. Pakzad, B. Nikolic, and V. Anantharam. High throughput low-density parity-check decoder architectures. in Proc. Global Conf. Commun., Nov.2001, pp. 3019-3024.
[34]D. Hocevar. A reduced complexity decoder architechture via layered decoding of LDPC codes. In2004 IEEE Workshop on Proc. Signal Processing Systems SIPS, Austiu, TX, USA, Oct.2004, pp.107-112.
[35]E. Sharon, S. Litsyn, and J. Goldberger. An efficient message passing schedule for LDPC decoding. In Proc.23rd IEEE Convention of Electrical and Electronics Engineers, Israel, Sept.2004, pp.223-226.
[36]H. Kfir and I. Kanter. Parallel versus sequential updating for belief propagation decoding. Physica A, Dec.2003,330:259-270.
[37]J. Zhang and M. Fossorier. Shuffled belief propagation decoding IEEE Trans, on Comm., Feb.2005,53:209-213.
[38]P. Radosavljevec, A. de Baynast, and J. R. Cavallaro. Optimized message passing schedules for LDPC decoding. In Proc. Thirty-Ninth Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA, Oct. 2005, pp.591-595
[39]A. I. Vila Casado, M. Griot, and R. D. Wesel. Informed Dynamic Scheduling for Belief-Propagation Decoding of LDPC Codes. In Proc. IEEE ICC 2007, Glasgow, Scotland, June,2007.
[40]A. I. Vila Casado, M. Griot, and R. D. Wesel. Improving LDPC Decoders via Informed Dynamic Scheduling. IEEE Information Theory Workshop 2007, Lake Tahoe, CA, September, 2007.
[41]T. Richardson. Error floors of LDPC codes. In Proc.41st Annual Allerton Conf. on Comm., Monticello, IL, 2003.
[43]Darabih and M. Fossorier. Improved min-sum decoding of LDPC codes using 2 dimensional normalization. IEEE Globalcom. 2006. Vol.7. pp.1187-1192.
[44]P. Radosavljevic, A. de Baynast, and J.R. Cavallaro. Optimized message passing schedules for LDPC decoding. In Proc. Thirty-Ninth Asilomar Conference on Signals, Systems and Computers, pages 591-595,2005.
[45]David J. C. MacKay and Matthew C. Davey. Two Small Gallager Codes. [online] available at http://wol.ra.phy.cam.ac.uk/mackay/.
[46]S.-Y. Chung, J. G. D. Forney, T. Richardson, and R. Urbanke. On the design of low-density parity-check codes within 0.0045dB of the Shannon limint. IEEE Communication. Letters, Feb.2001, vol.5,58-60.
[47]Gal Elidan, Ian McGraw, Daphne Koller. Residual Belief Propagation:Informed Scheduling for Asynchronous Message Passing. [online] available at http://www.robotics.stanford.edu/～galel/papers/ElidanRBP.pdf.
[48]V. Casado, A. I and M. Griot. Improving LDPC decoders via informed dynamic scheduling. IEEE Information. Theory. Workshop.2007. Vol.47. pp.800-803.
[49]Rovini, M. Rossi and P. Linsalata. Layered decoding of no-layered LDPC codes. Proc. 9th EUROMICRO. Conf.2006. Vol.15. pp.537 - 544.
[50]巩义.基于置信传播的二进制LDPC码动态译码算法研究[博士论文].广州.中山大学.2011.pp.23-50.
[51]V.Casado, M. Griot and R. Wesel. Overcoming LDPC trapping sets with informed scheduling for Belief Propagation Decoding of LDPC Codes. In Information Theory and Apllications Workshop USCD. January.2007. Vol.8. pp.35-49.
[52]赵香琴.喷泉码及其在协作通信系统中的应用研究[学位论文].成都.西南交通大学.2009.pp.9-28.
[2]Proakis J.G, "Digital Communication",3rd edition, New Nork:Me Graw-Hill,1995.
[3]Stephen G Wilson. Digital Modulation and Coding. Prentice-Hall, 1996.
[4]T J Richardson and R L Urbanke. The Capacity of Low-density Parity-check Codes Under Message-passing Decoding. IEEE Trans. Information Theory, Vol.47, pp.599-618, Feb.2001
[5]R G Gallgager. Information Theory and Reliable Communication. Wiley New York, 1968.
[6]Xingkai Bao, Jing Li. Adaptive Network Coded Cooperation(ANCC) for Wireless Relay Networks: Matching Code-on-Graph with Network-on-Graph. IEEE Transactions on wireless communications, VOL.7, NO.2, February 2008.
[7]E K Hall, Stephen G Wilson. Design and Analysis of Turbo Codes on Rayleigh Fading Channel. IEEE Journal on Selected Areas in Communications, Vol.16, No.2, pp.160-174, Feb.1998.
[8]Hamming R. Error Detecting and Error Correcting Codes[J]. Bell System Technical Journal,1950,29(2):147-160.
[9]Viterbi A. Error Bounds for Convolutional Codes and An Asymptotically Optimum Decoding Algorithm [J]. IEEE Trans. Information on Theory, 1967,13(2):260-269.
[10]Goppa V. Algebraico-Geometric Codes [J]. Izvestiya:Mathematics,1983,21(1):75-91.
[11]Goppa V. Geometry and Codes [J]. D Reidel Pub Co, 1988.
[12]Berrou Claude. The Ten-Year-Old Turbo Codes are Entering into Service. IEEE Communications Magazine, 2003, (8):110-115.
[13]D J C MacKay and R N Neal. Near Shannon limit performance of low density parity check codes [J]. Electronic Letters, Aug.1996,32:1645 - 1646.
[14]R. Gallgager. Low-density parity-check codes. IRE Trans. Information Theory, pp.21-28, Jan.1962.
[15]R. M. Tanner. A recursive approach to low complexity codes. IEEE Trans. Information Theory, pp.533-547, Sept. 1981.
[16]D. MacKay and R. Neal. Good codes based on very sparse matrices. The Proceeding of the 5th IMA Conf. in Cryptography and Coding,, C. Boyd, Ed., Lecture Notes in Computer Science, pp.100-111, Berlin, Germany: Springer, 1995.
[17]D. MacKay. Good error correcting codes based on very sparse matrices. IEEE Trans. Information Theory, pp.399-431, March 1999.
[18]N. Alon and M. Luby. A linear time erasure-resilient code with nearly optimal recovery. IEEE Trans. Information Theory, pp.1732-1736, Nov.1996.
[19]S. Y. Chung, G. D. Forney, Jr., T. J. Richardson, and R. Urbanke. On the design of low-density parity check codes within 0.0045dB of the Shannon limit. IEEE Commun.. Letters,2001,5(2):58-60.
[20]Spielman D. A.. Finding Good LDPC Codes, http://spielman@math.mit.edu.
[21]IEEE Std 802.16e. Air interface for fixed and mobile broadband wireless access systems. IEEE P802.16e/D12 Draft. Oct 2005. pp.626-630.
[22]Luby, M.. LT codes. Foundations of Computer Science,2002. Proceedings, of The 43rd Annual IEEE Symposium on Foundations of Computer Science, pp.271-280, 2002.
[23]Shokrollahi, A..Raptor codes. IEEE Trans. Inform. Theory, vol.52, no.6, pp.2551-2567, June 2006.
[24]R.Palanki and J. S. Yedidia, "Rateless codes on Noisy Channels," Proc. of IEEE International Symposium on Information Theory, Jun.2004.
[25]Castura, J.; Mao, Y.. Rateless coding over fading channels. IEEE Communications Letters,, vol.10, no.1, pp.46-48, Jan 2006.
[26]O. Etesami and A. Shokrollahi. Raptor codes on binary memoryless symmetric channels. IEEE Trans. Inform. Theory, vol.52, no.5, pp2033-2051, May 2006.
[27]Robert J McEliece. The Theory of Information and Coding. Addison-Wesley, Reading, 1977.
[28]William E Ryan. An Introduction to LDPC Codes, Department of Electrical and Computer Engineering, The University of Arizona, Aug 2003
[29]文红,符初生,周亮LDPC码原理与应用电子科技大学出版社,2006年：1-44,55-63,97-112.
[30]D.J.C.Mackay. Good error-correcting codes based on very sparsed matrices, IEEETrans.Inform.Theory, Marl 999:399-433.
[31]D.Divsalar, H. Jin, and R. McEliece, Proc.36th Annual Allerton Conf. on Comm., Control, and Cmputingce. Coding theorems for turbo-like codes, pp.201-210, Sept.1998.
[32]Omid Etesami and Amin Shokrollahi, "Raptor codes on Binary Memoryless Symmetric Channels,", IEEE Trans. Inform. Theory, vol.52, no.5, pp.2033-2051, May.2006.
[33]E. Yeo, P. Pakzad, B. Nikolic, and V. Anantharam. High throughput low-density parity-check decoder architectures. in Proc. Global Conf. Commun., Nov.2001, pp. 3019-3024.
[34]D. Hocevar. A reduced complexity decoder architechture via layered decoding of LDPC codes. In2004 IEEE Workshop on Proc. Signal Processing Systems SIPS, Austiu, TX, USA, Oct.2004, pp.107-112.
[35]E. Sharon, S. Litsyn, and J. Goldberger. An efficient message passing schedule for LDPC decoding. In Proc.23rd IEEE Convention of Electrical and Electronics Engineers, Israel, Sept.2004, pp.223-226.
[36]H. Kfir and I. Kanter. Parallel versus sequential updating for belief propagation decoding. Physica A, Dec.2003,330:259-270.
[37]J. Zhang and M. Fossorier. Shuffled belief propagation decoding IEEE Trans, on Comm., Feb.2005,53:209-213.
[38]P. Radosavljevec, A. de Baynast, and J. R. Cavallaro. Optimized message passing schedules for LDPC decoding. In Proc. Thirty-Ninth Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA, Oct. 2005, pp.591-595
[39]A. I. Vila Casado, M. Griot, and R. D. Wesel. Informed Dynamic Scheduling for Belief-Propagation Decoding of LDPC Codes. In Proc. IEEE ICC 2007, Glasgow, Scotland, June,2007.
[40]A. I. Vila Casado, M. Griot, and R. D. Wesel. Improving LDPC Decoders via Informed Dynamic Scheduling. IEEE Information Theory Workshop 2007, Lake Tahoe, CA, September, 2007.
[41]T. Richardson. Error floors of LDPC codes. In Proc.41st Annual Allerton Conf. on Comm., Monticello, IL, 2003.
[43]Darabih and M. Fossorier. Improved min-sum decoding of LDPC codes using 2 dimensional normalization. IEEE Globalcom. 2006. Vol.7. pp.1187-1192.
[44]P. Radosavljevic, A. de Baynast, and J.R. Cavallaro. Optimized message passing schedules for LDPC decoding. In Proc. Thirty-Ninth Asilomar Conference on Signals, Systems and Computers, pages 591-595,2005.
[45]David J. C. MacKay and Matthew C. Davey. Two Small Gallager Codes. [online] available at http://wol.ra.phy.cam.ac.uk/mackay/.
[46]S.-Y. Chung, J. G. D. Forney, T. Richardson, and R. Urbanke. On the design of low-density parity-check codes within 0.0045dB of the Shannon limint. IEEE Communication. Letters, Feb.2001, vol.5,58-60.
[47]Gal Elidan, Ian McGraw, Daphne Koller. Residual Belief Propagation:Informed Scheduling for Asynchronous Message Passing. [online] available at http://www.robotics.stanford.edu/～galel/papers/ElidanRBP.pdf.
[48]V. Casado, A. I and M. Griot. Improving LDPC decoders via informed dynamic scheduling. IEEE Information. Theory. Workshop.2007. Vol.47. pp.800-803.
[49]Rovini, M. Rossi and P. Linsalata. Layered decoding of no-layered LDPC codes. Proc. 9th EUROMICRO. Conf.2006. Vol.15. pp.537 - 544.
[50]巩义.基于置信传播的二进制LDPC码动态译码算法研究[博士论文].广州.中山大学.2011.pp.23-50.
[51]V.Casado, M. Griot and R. Wesel. Overcoming LDPC trapping sets with informed scheduling for Belief Propagation Decoding of LDPC Codes. In Information Theory and Apllications Workshop USCD. January.2007. Vol.8. pp.35-49.
[52]赵香琴.喷泉码及其在协作通信系统中的应用研究[学位论文].成都.西南交通大学.2009.pp.9-28.