Extremality for Gallager's Reliability Function $E_0$

TL;DR

This paper investigates extremal properties of Gallager's $E_0$ function for binary channels, characterizing the maximal and minimal $E_0( ho)$ curves among all binary DMCs passing through a specific point.

Contribution

It establishes extremality results for the $E_0( ho)$ curves of BEC and BSC channels within a class of binary DMCs constrained by a point on their $E_0( ho)$ curve.

Findings

Binary erasure channel maximizes $E_0( ho)$ curve.

Binary symmetric channel minimizes $E_0( ho)$ curve.

Results provide bounds for $E_0( ho)$ among binary DMCs.

Abstract

We describe certain extremalities for Gallager's $E_0$ function evaluated under the uniform input distribution for binary input discrete memoryless channels. The results characterize the extremality of the $E_0(\rho)$ curves of the binary erasure channel and the binary symmetric channel among all the $E_0(\rho)$ curves that can be generated by the class of binary discrete memoryless channels whose $E_0(\rho)$ curves pass through a given point $(\rho_0, e_0)$, for some $\rho_0 > -1$.