标题： 一个关于噪声函数$\ell_q$范数的改进界 显示英文标题

标题： An improved bound on $\ell_q$ norms of noisy functions

摘要： 设$T_{\epsilon}$，$0 \le \epsilon \le 1/2$是作用在布尔立方体$\{0,1\}^n$上的噪声算子。 设$f$是$\{0,1\}^n$上的非负函数，并设$q \ge 1$。 In arXiv:1809.09696 the $\ell_q$ norm of $T_{\epsilon} f$ was upperbounded by the average $\ell_q$ norm of conditional expectations of $f$, given sets whose elements are chosen at random with probability $\lambda$, depending on $q$ and on $\epsilon$. 在本文中，我们证明了对于整数$q \ge 2$的不等式，参数$\lambda$更小（更优）。新的不等式对于子立方体的特征函数是紧的。作为应用，根据 arXiv:2008.07236，我们表明若\[ R ~<~ 1 - \log_2\left(1 + \sqrt{4p(1-p)}\right). \]，则速率$R$的 Reed-Muller 码$C$在$\mathrm{BSC}(p)$上以高概率解码错误。这是对 arXiv:2008.07236 中估计的一个（次要）改进。 显示英文摘要

摘要： Let $T_{\epsilon}$, $0 \le \epsilon \le 1/2$, 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 $q \ge 1$. In arXiv:1809.09696 the $\ell_q$ norm of $T_{\epsilon} f$ was upperbounded by the average $\ell_q$ norm of conditional expectations of $f$, given sets whose elements are chosen at random with probability $\lambda$, depending on $q$ and on $\epsilon$. In this note we prove this inequality for integer $q \ge 2$ with a better (smaller) parameter $\lambda$. 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 $\mathrm{BSC}(p)$ with high probability if \[ R ~<~ 1 - \log_2\left(1 + \sqrt{4p(1-p)}\right). \] This is a (minor) improvement on the estimate in arXiv:2008.07236.