Extremal Channels of Gallager\´s $E_{0}$ Under the Basic Polarization Transformations

We study the extremality of the binary erasure channel and the binary symmetric channel for Gallager´s reliability function E₀ of binary input discrete memoryless channels evaluated under the uniform input distribution from the aspect of channel polarization. In particular, we show that amongst all binary discrete memoryless channels of a given E₀(ρ) value, for a fixed ρ ≥ 0, the binary erasure channel and the binary symmetric channel are extremal in the evolution of E₀ under the one-step polarization transformations.