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LDPC codes

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Codes and Turbo Codes

Part of the book series: Collection IRIS ((IRIS))

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Abstract

Low Density Parity Check (LDPC) codes make up a class of block codes that are characterized by a sparse parity check matrix. They were first described in Gallager’s thesis at the beginning of the 60s [9.21]. Apart from the hard input decoding of LDPC codes, this thesis proposed iterative decoding based on belief propagation (BP). This work was forgotten for 30 years. Only a few rare studies referred to it during this dormant period, in particular, Tanner’s which proposed a generalization of the Gallager codes and a bipartite graph [9.53] representation.

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Bibliography

  1. PB. Ammar, B. Honary, Y. Kou, and S. Lin. Construction of low density parity check codes: a combinatoric design approach. In Proceedings of IEEE International Symposium on Information Theory (ISIT′02), July 2002.

    Google Scholar 

  2. M. Ardakani, T.H. Chan, and F.R. Kschischang. Properties of the exit chart for one-dimensional ldpc decoding schemes. In Proceedings of Canadian Workshop on Information Theory, May 2003.

    Google Scholar 

  3. M. Ardakani and F.R. Kschischang. Designing irregular lpdc codes using exit charts based on message error rate. In Proceedings of International Symposium on Information Theory (ISIT′02), July 2002.

    Google Scholar 

  4. C. Berrou and S. Vaton. Computing the minimum distance of linear codes by the error impulse method. In Proceedings of IEEE International Symposium on Information Theory, July 2002.

    Google Scholar 

  5. C. Berrou, S. Vaton, M. Jézéquel, and C. Douillard. Computing the minimum distance of linear codes by the error impulse method. In Proceedings of IEEE Global Communication Conference (Globecom′2002), pages 1017–1020, Taipei, Taiwan, Nov. 2002.

    Google Scholar 

  6. M. Blaum, P. Farrel, and H. Van Tilborg. Chapter 22: Array codes. In Handbook of Coding Theory. Elsevier, 1998.

    Google Scholar 

  7. J.W. Bond, S. Hui, and H. Schmidt. Constructing low-density parity-check codes. EUROCOMM 2000. Information Systems for Enhanced Public Safety and Security, IEEE AFCEA, May 2000.

    Google Scholar 

  8. E. Boutillon, J. Castura, and F.R. Kschischang. Decoder-first code design. In Proceedings of 2nd International Symposium on Turbo Codes & Related Topics, pages 459–462, Brest, France, 2000.

    Google Scholar 

  9. J. Campello and D.S. Modha. Extended bit-filling and ldpc code design. Proceedings of IEEE Global Telecommunications Conference (GLOBE-COM′01), pages 985–989, Nov. 2001.

    Google Scholar 

  10. J. Chen and M. Fossorier. Near optimum universal belief propagation based decoding of low-density parity check codes. IEEE Transactions on Communications, 50:406–414, March 2002.

    Article  Google Scholar 

  11. J. Chen and M.P.C. Fossorier. Density evolution for two improved bpbased decoding algorithms of ldpc codes. IEEE Communications Letters, 6:208–210, May 2002.

    Article  Google Scholar 

  12. Y. Chen and D. Hocevar. A fpga and asic implementation of rate 1/2, 8088-b irregular low density parity check decoder. In Proceedings of IEEE Global Telecommunications Conference (GLOBECOM′03), 1–5 Dec. 2003.

    Google Scholar 

  13. S.-Y. Chung. On the Construction of some Capacity-Approaching Coding Schemes. PhD thesis, MIT, Cambridge, MA, 2000.

    Google Scholar 

  14. S.-Y. Chung, T.J. Richardson, and R.L. Urbanke. Analysis of sum-product decoding of low-density parity-check codes using a gaussian approximation. IEEE Transactions on Information Theory, 47, Feb. 2001.

    Google Scholar 

  15. D. Declercq. Optimisation et performances des codes ldpc pour des canaux non standards. Master’s thesis, Université de Cergy Pontoise, Dec. 2003.

    Google Scholar 

  16. D. Divsalar, H. Jin, and R. J. McEliece. Coding theorems for turbo-like codes. In Proceedings of 36th Allerton Conference on Communication, Control, and Computing, pages 201–210, Sept. 1998.

    Google Scholar 

  17. I.B. Djordjevic, S. Sankaranarayanan, and B.V. Vasic. Projective-plane iteratively decodable block codes for wdm high-speed long-haul transmission systems. Journal of Lightwave Technology, 22, March 2004.

    Google Scholar 

  18. E. Eleftheriou and S. Olcer. Low-density parity-check codes for digital subscriber lines. In Proceedings of International Conference on Communications, pages 1752–1757, 28 Apr.–2 May 2002.

    Google Scholar 

  19. J. L. Fan. Array codes as low-density parity-check codes. In Proceedings of 2nd Symposium on Turbo Codes, pages 543–546, Brest, France, Sept. 2000.

    Google Scholar 

  20. M.P.C. Fossorier, M. Mihaljevic, and I. Imai. Reduced complexity iterative decoding of low-density parity-check codes based on belief propagation. IEEE Transactions on Commununications, 47:673–680, May 1999.

    Article  Google Scholar 

  21. R. G. Gallager. Low-Density Parity-Check Codes. MIT Press, Cambridge, MA, 1963.

    Google Scholar 

  22. J. Garcia-Frias and Wei Zhong. Approaching shannon performance by iterative decoding of linear codes with low-density generator matrix. IEEE Communications Letters, 7:266–268, June 2003.

    Article  Google Scholar 

  23. F. Guilloud. Generic Architecture for LDPC Codes Decoding. PhD thesis, ENST Paris, July 2004.

    Google Scholar 

  24. F. Guilloud, E. Boutillon, and J.-L. Danger. Lambda-min decoding algorithm of regular and irregular ldpc codes. In Proceedin gs of 3rd International Symposium on Turbo Codes & Related Topics, 1–5 Sept. 2003.

    Google Scholar 

  25. D. Haley, A. Grant, and J. Buetefuer. Iterative encoding of low-density parity-check codes. In Proceedings of IEEE Global Telecommunications Conference (GLOBECOM′02), Nov. 2002.

    Google Scholar 

  26. X.-Y. Hu, M.P.C. Fossorier, and E. Eleftheriou. On the computation of the minimum distance of low-density parity-check codes. In Proceedings of IEEE International Conference on Communications (ICC′04), 2004.

    Google Scholar 

  27. X.-Y. Hu and R. Mittelholzer. An ordered-statistics-based approximation of the sum-product algorithm. In Proceedings of IEEE International Telecommunications Symposium, Natal, Brazil, 8–12 Sept. 2002.

    Google Scholar 

  28. X.-Y. Hu and R. Mittelholzer. A sorting-based approximation of the sum-product algorithm. Journal of the Brazilian Telecommunications Society, 18:54–60, June 2003.

    Google Scholar 

  29. X.Y. Hu, E. Eleftheriou, and D.-M. Arnold. Progressive edge-growth tanner graphs. In Proceedings of IEEE Global Telecommunications Conference (GLOBECOM′01), Nov. 2001.

    Google Scholar 

  30. S.J. Johnson and S.R. Weller. Construction of low-density parity-check codes from kirkman triple systems. In Proceedings of IEEE Global Telecommunication Conference (GLOBECOM′01), volume 2, pages 970–974, 25–29 Nov. 2001.

    Google Scholar 

  31. S.J. Johnson and S.R. Weller. Regular low-density parity-check codes from combinatorial designs. In Proceedin gs of Information Theory Workshop, pages 90–92, Sept. 2001.

    Google Scholar 

  32. C. Jones, E. Vallés, M. Smith, and J. Villasenor. Approximate min* constraint node updating for ldpc code decoding. In Proceedings of IEEE Military Communications Conference (MILCOM′03), 13–16 Oct. 2003.

    Google Scholar 

  33. R. Koetter and P.O. Vontobel. Graph-covers and the iterative decoding of finite length codes. In Proceedings of 3rd International Symposium on turboCodes & Related Topics, Sept. 2003.

    Google Scholar 

  34. Y. Kou, S. Lin, and M.P.C. Fossorier. Low-density parity-check codes based on finite geometries: A rediscovery and new results. IEEE Transactions on Information Theory, 47:2711–2736, Nov. 2001.

    Article  MATH  MathSciNet  Google Scholar 

  35. F.R. Kschischang and B.J. Frey. Iterative decoding of compound codes by probability propagation in graphical models. IEEE Journal on Selected Areas in Commununications, 16:219–230, 1998.

    Article  Google Scholar 

  36. D. J. C. MacKay. Good error-correcting codes based on very sparse matrices. IEEE Transactions on Information Theory, 45(2):399–431, March 1999.

    Article  MATH  MathSciNet  Google Scholar 

  37. D. J. C. MacKay and M. C. Davey. Evaluation of gallager codes for short block length and high rate applications. In B. Marcus and J. Rosenthal, editors, Codes, Systems and Graphical Models, volume 123 of IMA Volumes in Mathematics and its Applications,, pages 113–130. Springer, New York, 2000.

    Google Scholar 

  38. David J.C. MacKay and Michael S. Postol. Weaknesses of margulis and ramanujan-margulis low-density parity-check codes. Electronic Notes in Theoretical Computer Science, 74, 2003.

    Google Scholar 

  39. D.J.C MacKay and R.M. Neal. Good codes based on very sparse matrices. In Proceedings of 5th IMA Conference on CryproGraphy and Coding, Berlin, Germany, 1995.

    Google Scholar 

  40. D.J.C MacKay, S.T. Wilson, and M.C. Davey. Comparison of constructions of irregular gallager codes. IEEE Transactions on Communications, 47:1449–1454, Oct. 1999.

    Article  Google Scholar 

  41. M.M. Mansour and N.R. Shanbhag. Turbo decoder architectures for low-density parity-check codes. In Proceedings of IEEE Global Telecommunications Conference (GLOBECOM′02), 17–21 Nov. 2002.

    Google Scholar 

  42. Y. Mao and A.H. Banihashemi. Decoding low-density parity-check codes with probabilistic scheduling. IEEE Communications Letters, 5:414–416, Oct. 2001.

    Article  Google Scholar 

  43. G. A. Margulis. Explicit construction of graphs without short cycles and low density codes. Combinatorica, 2(1):71–78, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  44. R. J. McEliece, D. J. C. MacKay, and J.-F. Cheng. Turbo decoding as an instance of pearle’s belief propagation algorithm. IEEE Journal on Selected Areas in Commununications, 16:140–152, Feb. 1998.

    Article  Google Scholar 

  45. T.R. Oenning and Jaekyun Moon. A low-density generator matrix interpretation of parallel concatenated single bit parity codes. IEEE Transactions on Magnetics, 37:737–741, 2001.

    Article  Google Scholar 

  46. T. Okamura. Designing ldpc codes using cyclic shifts. In Proceedings of IEEE International Symposium on Information Theory (ISIT′03), July 2003.

    Google Scholar 

  47. A. Prabhakar and K. Narayanan. Pseudorandom construction of low density parity-check codes using linear congruential sequences. IEEE Transactions on Communications, 50:1389–1396, Sept. 2002.

    Article  Google Scholar 

  48. T.J. Richardson, M.A. Shokrollahi, and R.L. Urbanke. Design of capacity-approaching irregular low-density parity-check codes. IEEE Transactions on Information Theory, 47:619–637, Feb. 2001.

    Article  MATH  MathSciNet  Google Scholar 

  49. T.J. Richardson and R.L Urbanke. Efficient encoding of low-density parity-check codes. IEEE Transactions on Information Theory, 47:638–656, Feb. 2001.

    Article  MATH  MathSciNet  Google Scholar 

  50. J. Rosenthal and P. Vontobel. Constructions of ldpc codes using ramanujan graphs and ideas from margulis. In Proceedings of the 38-th Annual Allerton Conference on Communication, Control, and Computing, pages 248–257, 2000.

    Google Scholar 

  51. J. Rosenthal and P. Vontobel. Constructions of regular and irregular ldpc codes using ramanujan graphs and ideas from margulis. In Proceedings of IEEE International Symposium on Information Theory (ISIT′01), page 4, June 2001.

    Google Scholar 

  52. M. Sipser and D.A. Spielman. Expander codes. IEEE Transactions on Information Theory, 42:1710–1722, Nov. 1996.

    Article  MATH  MathSciNet  Google Scholar 

  53. R. M. Tanner. A recursive approach to low complexity codes. IEEE Transactions on Information Theory, IT-271:533–547, Sept. 1981.

    Article  MathSciNet  Google Scholar 

  54. R.M. Tanner. A [155, 64, 20] sparse graph (ldpc) code. In Proceedings of IEEE International Symposium on Information Theory, Sorrento, Italy, June 2000.

    Google Scholar 

  55. S. ten Brink. Convergence of iterative decoding. IEE Electronics Letters, 35:806–808, May 1999.

    Article  Google Scholar 

  56. S. ten Brink. Iterative decoding trajectories of parallel concatenated codes. In Proceedings of 3rd IEEE ITG Conference on Source and Channel Coding, pages 75–80, Munich, Germany, Jan. 2000.

    Google Scholar 

  57. T. Tian, C. Jones, J.D. Villasenor, and R.D. Wesel. Construction of irregular ldpc codes with low error floors. In Proceedings of IEEE International Conference on Communications (ICC′03), 2003.

    Google Scholar 

  58. B. Vasic. Combinatorial constructions of low-density parity check codes for iterative decoding. In Proceedings of IEEE International Symposium on Information Theory (ISIT′02), July 2002.

    Google Scholar 

  59. B. Vasic. High-rate low-density parity check codes based on anti-pasch affine geometries. In Proceedings of IEEE International Conference on Communications (ICC′02), volume 3, pages 1332–1336, 2002.

    Google Scholar 

  60. B. Vasic, E.M. Kurtas, and A.V. Kuznetsov. Kirkman systems and their application in perpendicular magnetic recording. IEEE Transactions on Magnetics, 38:1705–1710, July 2002.

    Article  Google Scholar 

  61. B. Vasic, E.M. Kurtas, and A.V. Kuznetsov. Ldpc codes based on mutually orthogonal latin rectangles and their application in perpendicular magnetic recording. IEEE Transactions on Magnetics, 38:2346–2348, Sept. 2002.

    Article  Google Scholar 

  62. F. Verdier and D. Declercq. A ldpc parity check matrix construction for parallel hardware decoding. In Proceedings of 3rd International Symposium on Turbo Codes & related topics, 1–5 Sept. 2003.

    Google Scholar 

  63. Eric W. Weisstein. Weisstein. from Mathworld, http://mathworld.wolfram.com/BlockDesign.html.

  64. N. Wiberg. Codes and Decoding on General Graphs. PhD thesis, Linköping University, 1996.

    Google Scholar 

  65. X.-Y.Hu, E. Eleftheriou, D.-M. Arnold, and A. Dholakia. Efficient implementations of the sum-product algorithm for decoding ldpc codes. In Proceedings of IEEE Global Telecommunications Conference (GLOBE-COM′01), pages 1036–1036, Nov. 2001.

    Google Scholar 

  66. E. Yeo, B. Nikolic, and V. Anantharam. High throughput low-density parity-check decoder architectures. In Proceedings of IEEE Global Telecommunications Conference (GLOBECOM′01), San Antonio, 25–29 Nov. 2001.

    Google Scholar 

  67. H. Zhang and J.M.F. Moura. The design of structured regular ldpc codes with large girth. In Proceedings of IEEE Global Telecommu nicat ion s Conference (GLOBECOM′03), Dec. 2003.

    Google Scholar 

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Editors and Affiliations

  1. Télécom Bretagne, CS 83818, 29238, Brest Cedex 3, France

    Claude Berrou

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(2010). LDPC codes. In: Berrou, C. (eds) Codes and Turbo Codes. Collection IRIS. Springer, Paris. https://doi.org/10.1007/978-2-8178-0039-4_9

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  • DOI: https://doi.org/10.1007/978-2-8178-0039-4_9

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  • Print ISBN: 978-2-8178-0038-7

  • Online ISBN: 978-2-8178-0039-4

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