Lower bounds on the spectrum and error rate LDPC code ensembles

We consider the ensemble of regular LDPC codes and obtain an expression for the second moment of the distance spectrum. We show how this expression can be used to derive a lower bound on the probability that the growth rate of a randomly chosen code from the ensemble is equal to the growth rate of the average distance spectrum, when the block length is sufficiently large. In particular, when the connectivity of the code is sufficiently large, the distance spectrum of a code in the ensemble is concentrated. We then derive a lower bound on the probability (confidence level) that the minimum distance and error rate, respectively, of a randomly chosen code from the ensemble are upper and lower bounded by some values (which depend on the confidence level)