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

TL;DR

This paper generalizes Samorodnitsky's noisy function inequality to broader settings, enabling new applications in error-correcting codes over arbitrary finite alphabets and improving understanding of code performance on various channels.

Contribution

The authors extend Samorodnitsky's inequality to functions over arbitrary product spaces and determine the optimal parameters for all relevant cases, broadening its applicability.

Findings

Generalized the inequality to functions on arbitrary product spaces.

Determined optimal parameters for the inequality for all relevant q, μ, ρ.

Applied the results to show coding guarantees over any finite alphabet.

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.