Thresholds for turbo codes

We prove the existence of thresholds for turbo codes and we prove concentration of the performance of turbo codes within the ensemble determined by the random interleaver. In effect, we show that the results obtained for low-density parity-check codes extend to turbo codes. The main technical innovation is to rigorously show that dependence of output extrinsic information on input priors decays with distance along the trellis. In an infinitely long turbo code the densities of the extrinsic information fulfill a certain symmetry condition which we call the consistency condition. This condition provides the basis for an efficient Monte-Carlo algorithm for the determination of thresholds for turbo codes. Thresholds of all symmetric parallel concatenated codes of memory up to 6 have been determined