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

TL;DR

This paper derives the asymptotic behavior of the bit error probability for irregular expurgated LDPC codes over BEC after a fixed number of iterations, extending previous results for regular ensembles.

Contribution

It provides a new analytical expression for the coefficient  that describes finite-length effects for irregular expurgated LDPC codes under iterative decoding.

Findings

Asymptotic  depends on ensemble, iterations, and erasure probability

Numerical estimates of  converge rapidly to the analytical result

Extension of regular ensemble analysis to irregular expurgated ensembles

Abstract

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.