On the Distribution of Low-Weight Codewords for Turbo Codes

We determine the generating function counting the asymptotic number of (minimal) codewords of low weight for standard parallel concatenated (turbo) code ensembles. We then show that the number of minimal codewords of weight up to some fixed constant follows in the limit of large block lengths a Poisson distribution. In analogy to the standard random graph model, we conjecture that this statement remains true as long as we look at codewords of weight O((logn)), where n is the blocklength. We then show that parallel concatenated turbo code ensembles have the same error-floor behavior as standard low-density parity-check ensembles LDPC(n,λ,ρ) with λ′(0)ρ′(1) > 1. Using this analogy we conjecture a compact form of the stability condition for parallel concatenated turbo code ensembles. This condition agrees with the few examples known in the literature. It has the pleasing interpretation to characterize that channel value at which the union bound applied to the asymptotic error floor expression diverges.