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 E-0 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 E-0(rho) value, for a fixed rho >= 0, the binary erasure channel and the binary symmetric channel are extremal in the evolution of E-0 under the one-step polarization transformations.