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.