Generalized Samorodnitsky noisy function inequalities, with applications to error-correcting codes

Abstract

An inequality by Samorodnitsky states that if $f : \mathbb{F}_2^n \to \mathbb{R}$ is a nonnegative boolean function, and $S \subseteq [n]$ is chosen by randomly including each coordinate with probability a certain $\lambda = \lambda(q,\rho) < 1$, then \begin{equation} \log \|T_\rho f\|_q \leq \mathbb{E}_{S} \log \|\mathbb{E}(f|S)\|_q\;. \end{equation} Samorodnitsky's inequality has several applications to the theory of error-correcting codes. Perhaps most notably, it can be used to show that \emph{any} binary linear code (with minimum distance $\omega(\log n)$) that has vanishing decoding error probability on the BEC$(\lambda)$ (binary erasure channel) also has vanishing decoding error on \emph{all} memoryless symmetric channels with capacity above some $C = C(\lambda)$. Samorodnitsky determined the optimal $\lambda = \lambda(q,\rho)$ for his inequality in the case that $q \geq 2$ is an integer. In this work, we generalize the inequality to $f : \Omega^n \to \mathbb{R}$ under any product probability distribution $\mu^{\otimes n}$ on $\Omega^n$; moreover, we determine the optimal value of $\lambda = \lambda(q,\mu,\rho)$ for any real $q \in [2,\infty]$, $\rho \in [0,1]$, and distribution~$\mu$. As one consequence, we obtain the aforementioned coding theory result for linear codes over \emph{any} finite alphabet.