On the Evaluation of the Polyanskiy-Poor–Verdú Converse Bound for Finite Block-Length Coding in AWGN

A tight converse bound to the channel coding rate in the finite block-length regime and under additive white Gaussian noise conditions was recently proposed by Polyanskiy-Poor-Verdú (PPV). The bound is a generalization of a number of other classical results, and it was also claimed to be equivalent to Shannon's 1959 cone packing bound. In this paper, we investigate methods for a reliable numerical evaluation of the bound, which is troublesome even for not too large values of the block-length n, by compactly expressing the Polyanskiy, Poor, and Verdú (PPV) converse bound in terms of non-central chi-squared distributions, and by evaluating those through an integral expression and a corresponding series expansion which exploit a method proposed by Temme. As a result, a robust evaluation method and new insights on the bound's asymptotics, as well as new approximate expressions, are obtained.