Polarization Improves

We prove that channel combining and splitting via Arikan´s polarization transformation improves Gallager´s reliability function E₀ for binary input channels. In this sense, polarization creates E₀. This observation gives yet another justification as to why the polar transform yields capacity achieving and low complexity codes: the improvement in E₀ translates to an improvement in complexity-error-probability trade-off. In analyzing polar codes, one examines auxiliary random processes that follow the evolution of information measures as an underlying communication channel undergoes a sequence of transformations. The conclusion of this paper shows that the E₀ process associated to such an analysis is a submartingale.