Performance analysis of linear codes under maximum-likelihood decoding: a tutorial
Communications and Information Theory
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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 parameters on the overall system performance as well as a measure of goodness of the suboptimum decoding methods used in practice. 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, we develop a general bound referred to as the generalized TSB (GTSB). 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 Gallager's thesis in the early 1960s. For the random coding case, Gallager provided the optimal solution in a closed form while for the nonrandom case the problem has been an active area of research in information theory for many years. We prove that for a sphere code, the optimal HSR within the proposed GTSB is a hyper-cone. This will climax to the TSB of Poltyrev, one of the tightest bounds ever developed for binary block codes, and therefore terminates the search for a better Gallager region in the groundwork of the GTSB.