Finite-Length Analysis of Irregular Expurgated LDPC Codes under Finite Number of Iterations

Communication over the binary erasure channel (BEC) using low-density parity-check (LDPC) codes and belief propagation (BP) decoding is considered. The average bit error probability of an irregular LDPC code ensemble after a fixed number of iterations converges to a limit, which is calculated via density evolution, as the blocklength $n$ tends to infinity. The difference between the bit error probability with blocklength $n$ and the large-blocklength limit behaves asymptotically like $\alpha/n$, where the coefficient $\alpha$ depends on the ensemble, the number of iterations and the erasure probability of the BEC\null. In [1], $\alpha$ is calculated for regular ensembles. In this paper, $\alpha$ for irregular expurgated ensembles is derived. It is demonstrated that convergence of numerical estimates of $\alpha$ to the analytic result is significantly fast for irregular unexpurgated ensembles.