Variations on the Gallager bounds, connections, and applications

In recent years there has been renewed interest in deriving tight bounds on the error performance of specific codes and ensembles, based on their distance spectrum. In this paper, we discuss many reported upper bounds on the maximum likelihood decoding error probability and demonstrate the underlying connections that exist between them. In addressing the Gallager bounds and their variations, we focus on the Duman and Salehi variation, which originates from the standard Gallager bound. A large class of efficient recent bounds (or their Chernoff versions) is demonstrated to be a special case of the generalized second version of the Duman and Salehi bounds. Implications and applications of these observations are pointed out, including the fully interleaved fading channel, resorting to either matched or mismatched decoding. The proposed approach can be generalized to geometrically uniform non-binary codes, finite state channels, bit interleaved coded modulation systems, and it can be also used for the derivation of upper bounds on the conditional decoding error probability.