Luby et al. derived evolution of degree distributions in residual graphs for irregular LDPC code ensembles. Evolution of degree distributions in residual graphs is important characteristic which is used for finite-length analysis of the expected block and bit error probability over the binary erasure channel. In this paper, we derive detailed evolution of degree distributions in residual graphs for irregular LDPC code ensembles with joint degree distributions.

In this paper, we provide explicit representations of average weight and stopping set distributions and asymptotic expressions of their exponent for detailedly represented irregular LDPC code ensembles. Further we present numerical examples which compare a detailedly represented irregular LDPC code ensemble with a conventional one with respect to both of weight and stopping set distributions.