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An Overview of the Isoperimetric Method in Coding Theory (Extended Abstract) [Invited Paper]

  • Jean-Pierre Tillich
  • Gilles Zémor
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1746)

Abstract

When decoding a threshold phenomenon is often observed: decoding deteriorates very suddenly around some critical value of the channel parameter. Threshold behaviour has been studied in many situations outside coding theory and a number of tools have been developped. One of those turns out to be particularly relevant to coding, namely the derivation of isoperimetric inequalities for product measures on Hamming spaces. we discuss this approach and derive consequences.

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References

  1. 1.
    Bobkov, S., Goetze, F. (1996) Discrete Isoperimetric and Poincaré-type inequalities. Technical report SFB 343 University of Bielefeld 96–086. ftp://ftp.mathematik.uni-bielefeld.de/pub/papers/sfb343/pr96086.ps.gz
  2. 2.
    Margulis, G. (1974) Probabilistic characteristics of graphs with large connectivity. Problemy Peredachi Informatsii. 10, 101–108MathSciNetzbMATHGoogle Scholar
  3. 3.
    Talagrand, M. (1993) Isoperimetry, logarithmic Sobolev inequalities on the discrete cube, and Margulis’ graph connectivity theorem. Geometric and Functional Analysis. 3, 295–314.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Tillich, J-P., Zémor, G. (1999) Discrete inequalities and the probability of a decoding error. Submitted to Combinatorics, Probability & Computing. http://www.infres.enst.fr/ zemor/isoperimetric.ps
  5. 5.
    Zémor, G. (1994) Threshold effects in codes. In Algebraic coding, Springer-Verlag, LNCS 781 278–286.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Jean-Pierre Tillich
    • 1
  • Gilles Zémor
    • 2
  1. 1.Université Paris-SudOrsayFrance
  2. 2.École Nationale Supérieure des TélécommunicationsParis 13France

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