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Codes from Expander Graphs

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5228))

Abstract

We survey some recent work on codes based on bipartite expander graphs. The code symbols are associated with the branches and the symbols connected to a given node are restricted to be codewords in certain constituent codes (e.g. Reed–Solomon codes). This class turn out to contain some exellent codes. We give results on the parameters of the codes and methods for their encoding. We also analyze the performance under iterative decoding, partly based on a result on cores in random graphs.

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References

  1. Zèmor, G.: On expander codes. IEEE Trans. Inform. Theory (Special Issue on Codes on Graphs and iterative Algorithms) 47, 835–837 (2001)

    MATH  Google Scholar 

  2. Barg, A., Zèmor, G.: Error exponents of expander codes. IEEE Trans. Inform. Theory 48, 1725–1729 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  3. Tanner, M.: A Recursive Approach to Low Complexity Codes. IEEE Trans. Inform. Theory 27, 533–547 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  4. Tanner, M.: Explicit Concentrators from Generalized N–Gons. SIAM J. Alg. Disc. Meth. 5(3), 287–293 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  5. Tanner, M.: Minimum-Distance Bounds by Graph Analysis. IEEE Trans. Inform. Theory 47, 808–821 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  6. Sipser, M., Spielman, D.A.: Expander Codes. IEEE Trans. Inform. Theory 42(6), 1710–1722 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  7. Roth, R.M.: Introduction to Coding Theory. Cambridge University Press, Cambridge (2006)

    MATH  Google Scholar 

  8. Janwa, H., Lal, A.K.: On Tanner Codes: Minimum Distance and Decoding. In: AAECC, vol. 13, pp. 335–347 (2003)

    Google Scholar 

  9. Roth, R.M., Skachek, V.: Improved Nearly-MDS Expander Codes. IEEE Trans. Inform. Theory 52(8), 3650–3661 (2006)

    Article  MathSciNet  Google Scholar 

  10. Feit, W., Higman, G.: The nonexistence of certain generalized polygons. J. Algebra 1, 114–131 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  11. van Maldeghem, H.: Generalized Polygons. Birkhäuser, Basel (1998)

    MATH  Google Scholar 

  12. McEliece, R., Swanson, L.: On the error probability for Reed-Solomon codes. IEEE Trans. Inform. Theory 32, 701–703 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  13. Pittel, B., Spencer, J., Wormald, N.: Sudden emergence of a giant k-core in a random graph. J. Comb. Theory, Series B 67, 11–151 (1996)

    MathSciNet  Google Scholar 

  14. Janson, S., Luczak, M.J.: A simple solution to the k-core problem. Random Structures Algorithms 30(1-2), 50–62 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  15. Høholdt, T., Justesen, J.: Graph codes with Reed-Solomon component codes. In: Proceedings ISIT 2006, Seattle, Washington, July 2006, pp. 2022–2026 (2006)

    Google Scholar 

  16. Høholdt, T., Justesen, J.: Iterative decoding of product codes and graph codes with Reed-Solomon component codes. In: Proceedings ITW 2007 (September 2007)

    Google Scholar 

  17. Høholdt, T., Justesen, J.: Graph Codes with Reed-Solomon Component Codes. In: IEEE Trans. Inform. Theory (March 2008) (submitted)

    Google Scholar 

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Ángela Barbero

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© 2008 Springer-Verlag Berlin Heidelberg

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Høholdt, T. (2008). Codes from Expander Graphs. In: Barbero, Á. (eds) Coding Theory and Applications. Lecture Notes in Computer Science, vol 5228. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-87448-5_9

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  • DOI: https://doi.org/10.1007/978-3-540-87448-5_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-87447-8

  • Online ISBN: 978-3-540-87448-5

  • eBook Packages: Computer ScienceComputer Science (R0)

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