The slope scaling parameter for general channels, decoders, and ensembles

Scaling laws are a powerful way to analyze the performance of moderately sized iteratively decoded sparse graph codes. Our aim is to provide an easily usable finite-length optimization tool that is applicable to the wide variety of channels, blocklengths, error probability requirements, and decoders that one encounters for practical systems. The tool is aimed at non-experts in the field, who need to quickly find code designs that are comparable with the best known codes available today but do not have the luxury of spending months in doing so. In previous work we have shown how to compute scaling parameters for transmission over the binary erasure channel, as well as general channels and general quantized message-passing decoders when applied to regular ensembles. In this paper we show how to compute the message variance for a fixed number of iterations for irregular low-density parity-check ensembles. From these calculations the basic scaling parameter alpha can be deduced by determining the leading term of the limiting expression when the number of iterations tends to infinity and the channel parameter approaches the density evolution threshold.