# Threshold effects in codes

## Abstract

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 *f*_{c}(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 *f*_{c}(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.

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.G. Margulis,
*Probabilistic characteristics of graphs with large connectivity*, Problemy Peredachi Informatsii, 10 (1974), pp. 101–108.Google Scholar - 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.G. Zémor and G. Cohen,
*The threshold probability of a code*. Submitted to IEEE Trans on Inf Theory.Google Scholar