An improved bound on ℓ_q norms of noisy functions
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Let T_ϵ, 0 ≤ϵ≤ 1/2, be the noise operator acting on functions on the boolean cube {0,1}^n. Let f be a nonnegative function on {0,1}^n and let q ≥ 1. In arXiv:1809.09696 the ℓ_q norm of T_ϵ f was upperbounded by the average ℓ_q norm of conditional expectations of f, given sets whose elements are chosen at random with probability λ, depending on q and on ϵ. In this note we prove this inequality for integer q ≥ 2 with a better (smaller) parameter λ. The new inequality is tight for characteristic functions of subcubes. As an application, following arXiv:2008.07236, we show that a Reed-Muller code C of rate R decodes errors on BSC(p) with high probability if R < 1 - log_2(1 + √(4p(1-p))). This is a (minor) improvement on the estimate in arXiv:2008.07236.
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