Extremality Properties for the Basic Polarization Transformations

TL;DR

This paper investigates how the binary erasure channel (BEC) and binary symmetric channel (BSC) exhibit extremal properties in the evolution of Gallager's reliability function during channel polarization, revealing their unique roles among binary input channels.

Contribution

It demonstrates that BEC and BSC are extremal channels in the evolution of $E_0$ under polarization, given a fixed $E_0( ho)$ value, for all $ ho \, \geq 0$.

Findings

BEC and BSC are extremal in $E_0$ evolution during polarization.

Among channels with the same $E_0( ho)$, BEC and BSC show extremal behavior.

The extremality holds for all $ ho \geq 0$.

Abstract

We study the extremality of the BEC and the BSC for Gallager's reliability function $E_0$ evaluated under the uniform input distribution for binary input DMCs from the aspect of channel polarization. In particular, we show that amongst all B-DMCs of a given $E_0(\rho)$ value, for a fixed $\rho \geq 0$, the BEC and BSC are extremal in the evolution of $E_0$ under the one-step polarization transformations.