Optimal Self-Dual Inequalities to Order Polarized BECs

Abstract

$1 - (1-x^M) ^ {2^M} > (1 - (1-x)^M) ^{2^M}$ is proved for all $x \in [0,1]$ and all $M > 1$. This confirms a conjecture about polar code, made by Wu and Siegel in 2019, that $W^{0^m 1^M}$ is more reliable than $W^{1^m 0^M}$, where $W$ is any binary erasure channel and $M = 2^m$. The proof relies on a remarkable relaxation that $m$ needs not be an integer, a cleverly crafted hexavariate ordinary differential equation, and a genius generalization of Green's theorem that concerns function composition. The resulting inequality is optimal, $M$ cannot be $2^m - 1$, witnessing how far polar code deviates from Reed--Muller code.