Finite Geometry Low Density Parity-Check Codes: Construction, Structure, and Decoding
This paper present an algebraic and systematic method for constructing low density parity-check (LDPC) codes from a geometric point of view. LDPC codes are constructed based on the lines and points of a finite geometry. Two classes of LDPC codes are constructed based on the well known Euclidean and projective geometries over finite fields. Codes of these two classes are cyclic and they have good minimum distances and structural properties. They can be decoded in various ways and perform very well with the iterative decoding based on belief propagation (IDBP). These finite geometry LDPC codes can be extended and shortened in various ways. Extension by splitting columns of the parity-check matrices of these codes results in good extended finite geometry LDPC codes. Long extended finite geometry LDPC codes have been constructed and they achieve an error performance only a few tenths of a dB away from the Shannon limit with IDBP. Finite geometry LDPC codes are strong competitors to turbo codes for error control in communication and digital data storage systems.
Five different decoding algorithms are described in this paper and they range from low to high decoding complexity and from reasonably good to very good decoding performance.
KeywordsEntropy Propa Expense
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- R. G. Gallager, Low Density Parity-Check Codes, MIT Press, Cambridge, MA, 1963.Google Scholar
- M. C. Davey and D. J. C. MacKay, “Low Density Parity Check Codes over GF(q),” IEEE Communications Letters, June 1998.Google Scholar
- M. G. Luby, M. Mitzenmacher, M. A. Shokrollahi, and D. A. Spielman, “Improved Low-Density Parity-Check Codes Using Irregular Graphs and Belief Propagation,” Proceedings of 1998 IEEE International Symposium on Information Theory, p. 171, Cambridge, MA, Aug 1998.Google Scholar
- D. J. C. Mackay, “Gallager Codes that are Better Than Turbo Codes,” Proceedings of the 36th Allerton Conference on Communication, Control, and Computing, Monticello, IL, Sep. 1998.Google Scholar
- D. J. C. Mackay, “Sparce Graph Codes,” Proceedings of the 5th International Symposium on Communication Theory and Applications, Ambleside, UK, pp. 2–4, July 1999.Google Scholar
- Y. Kou, S. Lin and M. Fossorier, “Low Density Parity-Check Codes based on Finite Geometries: A Rediscovery,” Proceedings of the IEEE International Symposium on Information Theory, Sorrento, Italy, June 2000.Google Scholar
- S. Lin and Y. Kou, “A Geometric Approach to the Construction of Low Density Parity-Check Codes,” presented at the IEEE 29th Communication Theory Workshop, Haines City, FL, May 2000.Google Scholar
- J. Pearl, Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference, Morgan Kaufmann, San Mateo, 1988.Google Scholar
- R. Lucas, M. Fossorier, Y. Kou, and S. Lin, “Iterative Decoding of One-Step Majority Logic Decodable Codes Based on Belief Propagation,” accepted by IEEE Transactions on Communications, 1999.Google Scholar
- C. Berrou, A. Glavieux, and P. Thitimajshima, “Near Shannon Limit Error-Correcting Coding and Decoding: Turbo Codes,” Proc. 1993 IEEE International Conference on Communications, Geneva, Switzerland, pp. 1064–1070, May 1993.Google Scholar
- S. Lin and D. J. Costello Jr., Error Control Coding: Fundamentals and Applications, Prentice Hall, Englewood Cliffs, NJ, 1983.Google Scholar
- J. L. Massey, Threshold Decoding, MIT Press, Cambridge, MA 1963.Google Scholar
- N. Wiberg, “Codes and Decoding on General Graphs,” Ph.D Dissertation, Department of Electrical Engineering, University of Linköping, Linköping, Sweden, April 1996Google Scholar
- G. D. Forney, Jr., “Codes on Graphs: A Survey for Algebraists,” Proc. 13-th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Honolulu, HI, Nov. 1999.Google Scholar
- G. D. Forney, Jr., “Codes on Graphs: Normal Realizations,” submitted to IEEE Transactions on Information Theory, 1998.Google Scholar
- E. J. Weldon, Jr., “Euclidean Geometry Cyclic Codes,” Proceedings of the Symp. Combinatorial Math, University of North Carolina, Chapel Hill, NC, 1967.Google Scholar
- S. Lin, “On a Class of Cyclic Codes,” Error Correcting Codes, (Edited by H. B. Mann), John Wiley & Sons, Inc., New York, 1968.Google Scholar
- I. S. Rees, “A Class of Multiple-Error-Correcting Codes and the Decoding Scheme,” IRE Transactions, vol. IT-4, pp. 38–49, Sep. 1954.Google Scholar