On achieving an asymptotically error-free fixed-point of iterative decoding for perfect a priori information

In this paper we provide necessary and sufficient conditions for constituent codes in (multiple) concatenated and graph-based coding schemes to achieve an asymptotically errorfree iterative decoding fixed-point if the maximum possible a priori information is available. At least one constituent code in an iterative decoding scheme must satisfy these conditions in order to ensure an asymptotically vanishing bit error probability at the convergence point of the decoder. Our results are proved for arbitrary binary-input symmetric memoryless channels (BISMCs) and thus can be universally applied to many transmission scenarios. Specifically, using a factor graph framework, it is shown that non-inner codes in a serial concatenation or check nodes in generalized LDPC codes achieve perfect extrinsic information if and only if the minimum Hamming distance between codewords is two or greater. For inner codes in a serial concatenation, constituent codes in a parallel concatenation, or variable nodes in doubly-generalized LDPC codes the corresponding encoder condition for acquiring perfect extrinsic information is an infinite codeword weight for a weight-one input sequence. For this case we provide a general proof which holds for all linear encoders and BISMCs. We also show that these results can improve the performance of concatenated coding schemes.