Optimal Self-Dual Inequalities to Order Polarized BECs

04/16/2023

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by   Ting-Chun Lin, et al.

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1 - (1-x^M) ^ 2^M > (1 - (1-x)^M) ^2^M is proved for all x ∈ [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.