Finite-length analysis of a capacity-achieving ensemble for the binary erasure channel

In this paper, we consider the finite-length performance of a capacity-achieving sequence of irregular repeat-accumulate (IRA) code ensembles. We focus on a sequence of bit-regular ensembles with degree 3 which was shown to achieve capacity with bounded complexity [Pfister, 2005]. To characterize how fast the block length of the code must grow with respect to the truncation point of the degree distribution (i.e., maximum check degree), we compute an upper bound on the average weight enumerator. Based on this analysis, we present a particular truncation sequence that could achieve a minimum distance which grows like n¹3/ even as the gap to capacity goes to zero. We also consider the performance of these codes in the waterfall region by extending the finite-length scaling law [Amraoui] from low-density parity-check codes to IRA codes. This shows that the performance near the iterative decoding threshold is well characterized by a suitably scaled Q-function for large enough block length. Numerical results are given for the scaling parameters of this ensemble sequence and for a few other IRA codes. Unfortunately, the simulation results for the capacity-achieving sequence start to match the scaling law only for very large block lengths.