The error probability of Maximum-Likelihood (ML) soft-decision decoded binary block codes rarely accepts nice closed forms. In addition, for long codes ML decoding becomes prohibitively complex. Nevertheless, bounds on the performance of ML decoded systems provide insight into the effect of system param eters on the overall system performance as well as a measure of efficiency of the suboptim um decoding methods used in practice. In this dissertation, using the so-called Gallager's first bounding technique (involving a so-called Gallager region) and within the framework of Tangential Sphere Bound (TSB) of Poltyrev. a general bound referred to as the Generalized Tangential Sphere Bound (GTSB) is developed. The Gallager region is chosen to be a general Hyper-Surface of Revolution (HSR) which is optimized to tighten the bound. The search for the optimal Gallager region is a classical problem dating back to G allager’s thesis in the early 1960’s. For the random coding case. Gallager provided the optimal solution in a closed form while for the non-random case the problem has been an active area of research in information theory for many years.

GTSB is applied to two problems. First, to the word error probability of BPSKmodulated binary block codes with ML decoding, and second, to the nonuniform signaling of binary block codes with Maximum-A-Posteriori (MAP) decoding. For both, the optimum HSR is obtained and it is noted that in the former case the resulting bound is the TSB of Poltyrev which has been the tightest bound developed for binary block codes to date. This term inates the search for a better Gallager region in the groundwork of the GTSB for the underlying application.

In the development of TSB, union bound which is the simplest inequality from the large class of so-called Bonferroni-type inequalities in probability theory is utilized. In this work, first, a new 2nd-order Bonferroni-type inequality is proposed, and then, a new upper bound on the word error probability of sphere codes is developed within the framework of Gallager’s First Bounding Technique (GFBT) and the new Bonferroni-type inequality. The resulting bound in its first version is rather complicated as it requires the global geometrical properties of the code.

Subsequently, a second version of the above upper bound, referred to as the Added-Hyper-Plane (AHP) bound, is presented. This bound is very simple as it only requires the spectrum of the code. For some long binary block codes, improvements over the TSB of Poltyrev are reported;​ making the bound the new tightest bound on the performance of ML-decoded binary block codes.