On the Error-Correcting Capabilities of Cycle Codes of Graphs

We are interested in a function f(p) that represents the probability that a random subset of edges of a Δ-regular graph G contains half the edges of some cycle of G. f(p) is also the probability that a codeword is corrupted beyond recognition when words of the cycle code of G are submitted to the binary symmetric channel. We derive a precise upper bound on the largest p for which f(p) can vanish when the number of edges of G goes to infinity. To this end, we introduce the notion of fractional percolation on trees, and calculate the related critical probabilities.