On Finite-Dimensional Bounds for LDPC-like Codes with Iterative Decoding

This paper focuses on finite-dimensional upper and lower bounds on decodable thresholds assuming iterative decoding. Two noise measures will be considered: the Bhattacharyya noise parameter, and the expected soft bit of the considered channel. An iterative m-dimensional lower bound is derived for Zm LDPC codes in symmetric channels, which gives a sufficient stability condition for Zm LDPC codes and will be complemented by a new necessary stability condition introduced herein. Two new lower bounds are provided for binary-input/symmetric-output channels, including a two-dimensional iterative bound and a onedimensional non-iterative bound, the latter of which is the best known bound that is tight for BSCs. A reverse channel perspective is then used to generalize existing bounds from binary-input/symmetric-output channels to binary-input/non-symmetric-output channels.