Asymptotic Rate versus Design Rate

The rate of a code is one of its most important parameters. We consider sparse graph codes and ask whether the rate of a random element of an ensemble is typically close to the design rate of the ensemble. For regular LDPC ensembles this question was answered in the affirmative in (Miller and Cohen, 2003). We start by giving an alternative proof of this statement. We then show that essentially the same type of argument applies not only to regular ensembles but also to ensembles that are derived from regular ensembles in the sense that their degree distribution is the result of applying the peeling decoder to a regular code. As an immediate consequence we prove that for regular ensembles the asymptotic MAP EXIT value coincides with the asymptotic BP EXIT value. We then give a systematic construction of ensembles for which rate and design rate differ. To accomplish this, we first show that the duality theorem (Ashikhminet al., 2004) implies that the asymptotic BP EXIT and the MAP EXIT functions are identical for any channel parameter for which the density evolution (DE) equations have a unique fixed point.