Finite-Dimensional Bounds on ${\BBZ}_m$ and Binary LDPC Codes With Belief Propagation Decoders

This paper focuses on finite-dimensional upper and lower bounds on decodable thresholds of Zopf <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sub> and binary low-density parity-check (LDPC) codes, assuming belief propagation decoding on memoryless channels. A concrete framework is presented, admitting systematic searches for new bounds. Two noise measures are considered: the Bhattacharyya noise parameter and the soft bit value for a maximum a posteriori probability (MAP) decoder on the uncoded channel. For Zopf <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sub> LDPC codes, an iterative m-dimensional bound is derived for m-ary-input/symmetric-output channels, which gives a sufficient stability condition for Zopf <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sub> LDPC codes and is complemented by a matched necessary stability condition introduced herein. Applications to coded modulation and to codes with nonequiprobably distributed codewords are also discussed. For binary codes, two new lower bounds are provided for symmetric channels, including a two-dimensional iterative bound and a one-dimensional noniterative bound, the latter of which is the best known bound that is tight for binary-symmetric channels (BSCs), and is a strict improvement over the existing bound derived by the channel degradation argument. By adopting the reverse channel perspective, upper and lower bounds on the decodable Bhattacharyya noise parameter are derived for nonsymmetric channels, which coincides with the existing bound for symmetric channels