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

Abstract

A tight converse bound to channel coding rate in the finite block-length regime and under AWGN conditions was recently proposed by Polyanskiy, Poor, and Verdu (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. Unfortunately, its numerical evaluation is troublesome even for not too large values of the block-length n. In this paper we tackle the numerical evaluation by compactly expressing the PPV converse bound in terms of non-central chi-squared distributions, and by evaluating those through a 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 given.