In this paper, we conclude five kinds of methods for construction of the regular low - density parity matrix H and three kinds of methods for the construction of irregular low - density parity - check matrix H . Through the analy sis of the code rate and parameters of these eight kinds of structures, we find that the construction of low - density parity - check matrix tends to be more flexible a nd the parameter variability is enhanced. We propose that the current development cost should be lower with the progress of electronic technology and we need research on more practical Low - Density Parity - Check Codes (LDPC). Combined with the application of the quantum distribution key, we urgently need to explore the research direction of relevant theories and technologies of LDPC codes in other fields of quantum information in the future.
Gallager, R.G. (1963) Low-Density Parity-Check Codes. MIT Press, Cambridge. https://doi.org/10.7551/mitpress/4347.001.0001
Shannon, C.E. (1957) Certain Results in Coding Theory for Noisy Channels. Information and Control, 1, 6-25. https://doi.org/10.1016/S0019-9958(57)90039-6
MacKay, D.J.C. and Neal, R.M. (1995) Good Codes Based on Very Sparse Matrices, IMA International Conference on Cryptography & Coding, Cirencester, 17-19 December 1995, 100-111. https://doi.org/10.1007/3-540-60693-9_13
Soleymani, M.R., Gao, Y. and Vilaipornsawai, U. (2002) Low Density Parity Check Codes. In: Turbo Coding for Satellite and Wireless Communications, Springer, Boston, 177-194. https://doi.org/10.1007/0-306-47677-0_9
Richardson, T.J. and Urbanke, R.L. (2001) The Capacity of Low-Density Parity-Check Codes Under Message-Passing Decoding. IEEE Transactions on Information Theory, 47, 599-618. https://doi.org/10.1109/18.910577
Eroz, M. and Lee, L. (2005) Structured Low-Density Parity-Check Code Design for Next Generation Digital Video Broadcast. MILCOM 2005—2005 IEEE Military Communications Conference, Vol. 4, Atlantic, 17-20 October 2005, 2461-2466. https://doi.org/10.1109/MILCOM.2005.1606037
Purnamasari, R., Wijanto, H. and Hidayat, I. (2014) Design and Implementation of LDPC (Low Density Parity Check) Coding Technique on FPGA (Field Programmable Gate Array) for DVB-S2 (Digital Video Broadcasting-Satellite). 2014 IEEE International Conference on Aerospace Electronics and Remote Sensing Technology, Yogyakarta, 13-14 November 2014, 83-88. https://doi.org/10.1109/ICARES.2014.7024407
Van Nghia, T. (2016) Development of the Parallel BCH and LDPC Encoders Architecture for the Second Generation Digital Video Broadcasting Standards with Adjustable Encoding Parameters on FPGA, 2016 International Conference on Engineering and Telecommunication (EnT), Moscow, 29-30 November 2016, 104-109. https://doi.org/10.1109/EnT.2016.031
IEEE Std. 1890-2018 (2019) IEEE Standard for Error Correction Coding of Flash Memory Using Low-Density Parity-Check Codes. Institute of Electrical and Electronics Engineers, Manhattan, 1-51.
Wetcharungsri, J., Buabthong, N., Jantarachote, S., Sangwongngam, P. and Sripimanwat, K. (2013) Field-Programmable Gate Array Implementation of Low-Density Parity-Check Codes Decoder and Hardware Testbed. IEEE 2013 Tencon—Spring, Sydney, 17-19 April 2013, 104-107. https://doi.org/10.1109/TENCONSpring.2013.6584426
Bennett, C.H. and Brassard, G. (1984) Quantum Cryptography: Public Key Distribution and Coin Tossing. IEEE International Conference on Computers, Systems & Signal Processing, Bangalore, 9-12 December 1984, 175-179.
Zheng, X., Lau, F.C.M., Tse, C.K., He, Y. and Hau, S. (2009) Application of Complex-Network Theories to the Design of Short-Length Low-Density-Parity-Check Codes. Communications IET, 3, 1569-1577. https://doi.org/10.1049/iet-com.2008.0503
Chung, S.Y., Forney Jr., G.D., Richardson, T.J. and Urbanke, R. (2001) On the Design of Low-Density Parity-Check Codes within 0.0045 dB of the Shannon Limit. IEEE Communications Letters, 5, 58-60. https://doi.org/10.1109/4234.905935
Mahdi, A., Kanistras, N. and Paliouras, V. (2021) A Multirate Fully Parallel LDPC Encoder for the IEEE 802.11n/ac/ax QC-LDPC Codes Based on Reduced Complexity XOR Trees. IEEE Transactions on Very Large Scale Integration (VLSI) Systems, 29, 51-64. https://doi.org/10.1109/TVLSI.2020.3034046
Zhong, H. and Zhang, T. (2005) Block-LDPC: A Practical LDPC Coding System Design Approach. IEEE Transactions on Circuits and Systems I: Regular Papers, 52, 766-775. https://doi.org/10.1109/TCSI.2005.844113
Zhao, Y. and Lau, F.C.M. (2014) Implementation of Decoders for LDPC Block Codes and LDPC Convolutional Codes Based on GPUs. IEEE Transactions on Parallel and Distributed Systems, 25, 663-672. https://doi.org/10.1109/TPDS.2013.52
Amaricai, A., Stein, D. and Boncalo, O. (2020) Generalized Very High Throughput Unrolled LDPC Layered Decoder, 2020 28th Telecommunications Forum (TELFOR), Belgrade, 24-25 November 2020, 1-4. https://doi.org/10.1109/TELFOR51502.2020.9306537
Chan, C.H. and Lau, F.C.M. (2012) Parallel Decoding of LDPC Convolutional Codes Using OpenMP and GPU. 2012 IEEE Symposium on Computers and Communications (ISCC), Cappadocia, 1-4 July 2012, 225-227. https://doi.org/10.1109/ISCC.2012.6249298
Li, Y., Liu, B., Rong, B., Wu, Y., Gagnon, G., Gui, L., et al. (2013) Rate-Compatible LDPC-RS Product Codes Based on Raptor-Like LDPC Codes. 2013 IEEE International Symposium on Broadband Multimedia Systems and Broadcasting (BMSB), London, 5-7 June 2013, 1-6. https://doi.org/10.1109/BMSB.2013.6621783
Vafi, S. and Majid, N. (2016) Half Rate Quasi Cyclic Low Density Parity Check Codes Based on Combinatorial Designs. Journal of Computer and Communications, 4, 39-49. https://doi.org/10.4236/jcc.2016.412002
Dehghan, A. and Banihashemi, A.H. (2018) On the Tanner Graph Cycle Distribution of Random LDPC, Random Protograph-Based LDPC, and Random Quasi-Cyclic LDPC Code Ensembles. IEEE Transactions on Information Theory, 64, 4438-4451. https://doi.org/10.1109/TIT.2018.2805906
Mitchell, D.G.M., Lentmaier, M. and Costello, D.J. (2015) Spatially Coupled LDPC Codes Constructed from Protographs. IEEE Transactions on Information Theory, 61, 4866-4889. https://doi.org/10.1109/TIT.2015.2453267
Thorpe, J. (2003) Low-Density Parity-Check (LDPC) Codes Constructed from Protographs. Interplanetary Network Progress Report. Progress Report 42-154.
Gallager, R.G. (2015) Low-Density Parity-Check Codes. Springer India, New Delhi.
Gallager, R. (1962) Low-Density Parity-Check Codes. IRE Transactions on Information Theory, 8, 21-28. https://doi.org/10.1109/TIT.1962.1057683
Fan, J.L. (2000) Array Codes as Low-Density Parity-Check Codes. Proceedings of 2nd International Symposium on Turbo Codes and Related Topics, Brest, 543-546.
Honary, B., Lin, S., Gabidulin, E.M., Xu, J., Kou, Y., Moinian, A. and Ammar, B. (2004) On Construction of Low Density Parity Check Codes. 2nd International Workshop on Signal Processing for Wireless Communication (SPWC 2004), London, 11-26.
Ammer, B., Honary, B., Xu, Y. and Lin, S. (2004) Construction of Low-Density Parity-Check Codes on Balanced Incomplete Block Designs. IEEE Transactions on Information Theory, 50, 1257-1269. https://doi.org/10.1109/TIT.2004.828144
Miladinovic, N. and Fossorier, M. (2004) Systematic Recursive Construction of LPDC Codes. IEEE Communications Letters, 8, 302-304. https://doi.org/10.1109/LCOMM.2004.827431
Tanner, R.M., Sridhara, D., Sridharan, A., Fuja, T.E. and Costello, D.J. (2004) LDPC Block and Convolutional Codes Based on Circulant Matrices. IEEE Transactions on Information Theory, 50, 2966-2984. https://doi.org/10.1109/TIT.2004.838370
Hakimi, S.L. and Bredeson, J. (1969) Ternary Graph Theoretic Error-Correcting Codes. IEEE Transactions on Information Theory, 15, 435-436. https://doi.org/10.1109/TIT.1969.1054312
Zhang, K., Jiang, X.Q., Feng, Y., Qiu, R. and Bai, E. (2020) High Efficiency Continuous-Variable Quantum Key Distribution Based on Quasi-Cyclic LDPC Codes. 2020 5th International Conference on Communication, Image and Signal Processing (CCISP), Chengdu, 13-15 November 2020, 38-42. https://doi.org/10.1109/CCISP51026.2020.9273490
Ali, I., Lee, H., Hussain, A. and Kim, S. (2019) Protograph-Based Folded Spatially Coupled LDPC Codes for Burst Erasure Channels. IEEE Wireless Communications Letters, 8, 516-519. https://doi.org/10.1109/LWC.2018.2878562
Liu, K., El-Khamy, M. and Lee, J. (2016) Finite-Length Algebraic Spatially-Coupled Quasi-Cyclic LDPC Codes. IEEE Journal on Selected Areas in Communications, 34, 329-344. https://doi.org/10.1109/JSAC.2015.2504273
Kong, L., He, L., Chen, P., Han, G. and Fang, Y. (2015) Protograph-Based Quasi-Cyclic LDPC Coding for Ultrahigh Density Magnetic Recording Channels. IEEE Transactions on Magnetics, 51, 1-4. https://doi.org/10.1109/TMAG.2015.2437397
Kulkarni, V. and Sankar, K.J. (2015) Design of Structured Irregular LDPC Codes from Structured Regular LDPC Codes. Proceedings of the 2015 3rd International Conference on Computer, Communication, Control and Information Technology (C3IT), Hooghly, 7-8 February 2015, 1-4. https://doi.org/10.1109/C3IT.2015.7060128
Mankar, M.V., Patil, A. and Asutkar, G.M. (2014) Single Mode Quasi-Cyclic LDPC Decoder Using Modified Belief Propagation. 2014 International Conference on Communication and Signal Processing, Melmaruvathur, 3-5 April 2014, 862-866. https://doi.org/10.1109/ICCSP.2014.6949966
Pisek, E., Rajan, D. and Cleveland, J.R. (2015) Trellis-Based QC-LDPC Convolutional Codes Enabling Low Power Decoders. IEEE Transactions on Communications, 63, 1939-1951. https://doi.org/10.1109/TCOMM.2015.2424434
Esfahanizadeh, H., Ram, E. and Cassuto, Y. (2020) Spatially Coupled Codes with Sub-Block Locality: Joint Finite Length-Asymptotic Design Approach. 2020 IEEE International Symposium on Information Theory (ISIT), Los Angeles, 21-26 June 2020, 467-472. https://doi.org/10.1109/ISIT44484.2020.9174265
Ram, E. and Cassuto, Y. (2021) Spatially Coupled LDPC Codes with Sub-Block Locality. IEEE Transactions on Information Theory, 67, 2739-2757. https://doi.org/10.1109/TIT.2021.3059751
Elkouss, D., Martinez-Mateo, J. and Martin, V. (2011) Information Reconciliation for Quantum Key Distribution. Quantum Information and Computation, 11, 226-238. https://doi.org/10.26421/QIC11.3-4-3
Martinez-Mateo, J., Elkouss, D. and Martin, V. (2012) Blind Reconciliation. Quantum Information and Computation, 12, 791-812. https://doi.org/10.26421/QIC12.9-10-5
Martinez-Mateo, J., Elkouss, D. and Martin, V. (2013) Key Reconciliation for High Performance Quantum Key Distribution. Scientific Reports, 3, Article No. 1576. https://doi.org/10.1038/srep01576
Martinez-Mateo, J., Pacher, C., Peev, M., Ciurana, A. and Martin, V. (2015) Demystifying the Information Reconciliation Protocol Cascade. Quantum Information and Computation, 15, 453-477. https://doi.org/10.26421/QIC15.5-6-6
Pedersen, T.B. and Toyran, M. (2015) High Performance Information Reconciliation for QKD with CASCADE. Quantum Information and Computation, 15, 419-434. https://doi.org/10.26421/QIC15.5-6-4
Barnault, L. and Declercq, D. (2003) Fast Decoding Algorithm for LDPC over GF (2q). 2003 IEEE Information Theory Workshop, Paris, 31 March-4 April 2003, 70-73. https://doi.org/10.1109/ITW.2003.1216697
Davey, M.C. and MacKay, D. (1998) Low-Density Parity Check Codes over GF (q). IEEE Communications Letters, 2, 165-167. https://doi.org/10.1109/4234.681360
Declercq, D. and Fossorier, M. (2007) Decoding Algorithms for Nonbinary LDPC Codes over GF (q). IEEE Transactions on Communications, 55, 633-643. https://doi.org/10.1109/TCOMM.2007.894088
Pacher, C., Martinez-Mateo, J., Duhme, J., Gehring, T. and Furrer, F. (2016) Information Reconciliation for Continuous-Variable Quantum Key Distribution Using Non-Binary Low-Density Parity-Check Codes. arXiv: 1602.09140.
Al-Rubaye, G.A., Tsimenidis, C.C. and Johnston, M. (2015) Non-Binary LDPC Coded OFDM in Impulsive Power Line Channels. 2015 23rd European Signal Processing Conference (EUSIPCO), Nice, 31 August-4 September 2015, 1431-1435. https://doi.org/10.1109/EUSIPCO.2015.7362620
Gehring, T., Händchen, V., Duhme, J., Furrer, F., Franz, T., Pacher, C., et al. (2015) Implementation of Continuous-Variable Quantum Key Distribution with Composable and One-Sided-Device-Independent Security against Coherent Attacks. Nature Communications, 6, Article No. 8795. https://doi.org/10.1038/ncomms9795
Montorsi, G. (2012) Analog Digital Belief Propagation. IEEE Communications Letters, 16, 1106-1109. https://doi.org/10.1109/LCOMM.2012.020712.112133
Sayir, J. (2014) Non-Binary LDPC Decoding Using Truncated Messages in the Walsh Hadamard Domain. ISITA 2014, International Symposium on Information Theory and Its Applications, Melbourne, 26-29 October 2014, 16-20.
Voicila, A., Declercq, D., Verdier, F., Fossorier, M. and Urard, P. (2010) Low-Complexity Decoding for Non-Binary LDPC Codes in High Order Fields. IEEE Transactions on Communications, 58, 1365-1375. https://doi.org/10.1109/TCOMM.2010.05.070096
Hosseinidehaj, N. and Malaney, R. (2016) CV-QKD with Gaussian and Non-Gaussian Entangled States over Satellite-Based Channels. 2016 IEEE Global Communications Conference (GLOBECOM), Washington DC, 4-8 December 2016, 1-7. https://doi.org/10.1109/GLOCOM.2016.7841711
Shen, Y., Cao, L., Wang, X.Y., Zou, J., Luo, W., Wang, Y.X., et al. (2020) On-Chip Continuous-Variable Quantum Key Distribution (CV-QKD) and Homodyne Detection. 2020 Optical Fiber Communications Conference and Exhibition (OFC), San Diego, 8-12 March 2020, 1-3. https://doi.org/10.1364/OFC.2020.W2A.53
Jayasooriya, S., Shirvanimoghaddam, M., Ong, L., Lechner, G. and Johnson, S.J. (2016) A New Density Evolution Approximation for LDPC and Multi-Edge Type LDPC Codes. IEEE Transactions on Communications, 64, 4044-4056. https://doi.org/10.1109/TCOMM.2016.2600660
Leverrier, A. and Grangier, P. (2009) Unconditional Security Proof of Long-Distance Continuous-Variable Quantum Key Distribution with Discrete Modulation. Physical Review Letters, 102, Article ID: 180504. https://doi.org/10.1103/PhysRevLett.102.180504
Zhou, C., Wang, X.Y. and Zhang, Z.G. (2021) Rate Compatible Reconciliation for Continuous-Variable Quantum Key Distribution Using Raptor-Like LDPC Codes. Science China Physics, Mechanics & Astronomy, 64, Article No. 260311. https://doi.org/10.1007/s11433-021-1688-4
Wang, X., Zhang, Y., Yu, S., Xu, B., Li, Z. and Guo, H. (2017) Efficient Rate-Adaptive Reconciliation for Continuous-Variable Quantum Key Distribution. Quantum Information and Computation, 17, 1123-1134. https://doi.org/10.26421/QIC17.13-14-4
Rakita, A., Nikoli’c, N., Mildner, M., Matiasek, J. and Elbe-Bürger, A. (2020) Re-Epithelialization and Immune Cell Behaviour in an ex Vivo Human Skin Model. Scientific Reports, 10, Article No. 1. https://doi.org/10.1038/s41598-019-56847-4
Bonello, N., Chen, S. and Hanzo, L. (2011) Low-Density Parity-Check Codes and Their Rateless Relatives. IEEE Communications Surveys & Tutorials, 13, 3-26. https://doi.org/10.1109/SURV.2011.040410.00042
Yu, Y., Jia, Z., Tao, W. and Dong, S. (2016) LDPC Codes Optimization for Differential Encoded LDPC Coded Systems with Multiple Symbol Differential Detection. 2016 IEEE 5th Global Conference on Consumer Electronics, Kyoto, 11-14 October 2016, 1-2. https://doi.org/10.1109/GCCE.2016.7800541
Yang, S.S., Lu, Z.G. and Li, Y.M. (2020) High-Speed Post-Processing in Continuous-Variable Quantum Key Distribution Based on FPGA Implementation. Journal of Lightwave Technology, 38, 3935-3941. https://doi.org/10.1109/JLT.2020.2985408