Algebraic Coding 1993: Algebraic Coding pp 278-286 | Cite as

Threshold effects in codes

  • Gilles Zémor
Bounds for Codes
Part of the Lecture Notes in Computer Science book series (LNCS, volume 781)


A theorem of Margulis states the existence of a threshold phenomenon in the probability of disconnecting a graph, given that each of its edges is independently severed with some probability p. We show how this theorem can be reinterpreted in the coding context: in particular we study the probability fc(p) of residual error after maximum likelihood decoding, when we submit a linear code C to a binary symmetric channel with error probability p. We show that the function fc(p) displays a threshold behaviour i.e. jumps suddenly from almost zero to almost one, and how the acuteness of the threshold effect grows with the minimal distance of C. Similar results for the erasure channel are also discussed.


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  1. 1.
    G. Margulis, Probabilistic characteristics of graphs with large connectivity, Problemy Peredachi Informatsii, 10 (1974), pp. 101–108.Google Scholar
  2. 2.
    M. Talagrand, Isoperimetry, logarithmic Sobolev inequalities on the discrete cube, and Margulis' graph connectivity theorem, Geometric and Functional Analysis, 3 (1993), pp. 295–314.Google Scholar
  3. 3.
    G. Zémor and G. Cohen, The threshold probability of a code. Submitted to IEEE Trans on Inf Theory.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Gilles Zémor
    • 1
  1. 1.Dépt. RéseauxEcole Nationale Supérieure des TélécommunicationsParis Cedex 13France

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