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

2025-08-09

AI Summary

This work generalizes the Samorodnitsky inequality—originally restricted to Boolean functions under specific distributions—to arbitrary real-valued functions over finite alphabets and general product probability distributions. The key theoretical advance is the first exact characterization of the optimal constant $lambda$ for all real $q geq 2$, yielding a unified formulation valid over arbitrary finite fields and non-uniform product distributions. Methodologically, the proof integrates noise operator analysis, conditional expectation techniques, and information-theoretic tools. This extension significantly broadens the inequality’s applicability and foundational scope. In application, the generalized inequality precisely characterizes decoding error behavior of linear codes over the binary erasure channel (BEC) and, more generally, over symmetric channels. It establishes that if a linear code achieves asymptotically negligible error probability on the BEC, then it also achieves reliable communication over any symmetric channel whose capacity exceeds a well-defined threshold—thereby bridging combinatorial coding theory with probabilistic functional inequalities.

Technology Category

• Machine Learning: Information Theory
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

An inequality by Samorodnitsky states that if $f : mathbb{F}_2^n o mathbb{R}$ is a nonnegative boolean function, and $S subseteq [n]$ is chosen by randomly including each coordinate with probability a certain $λ= λ(q,ρ) < 1$, then egin{equation} log |T_ρf|_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 $ω(log n)$) that has vanishing decoding error probability on the BEC$(λ)$ (binary erasure channel) also has vanishing decoding error on emph{all} memoryless symmetric channels with capacity above some $C = C(λ)$. Samorodnitsky determined the optimal $λ= λ(q,ρ)$ for his inequality in the case that $q geq 2$ is an integer. In this work, we generalize the inequality to $f : Ω^n o mathbb{R}$ under any product probability distribution $μ^{otimes n}$ on $Ω^n$; moreover, we determine the optimal value of $λ= λ(q,μ,ρ)$ for any real $q in [2,infty]$, $ρin [0,1]$, and distribution~$μ$. As one consequence, we obtain the aforementioned coding theory result for linear codes over emph{any} finite alphabet.

Problem

• Generalize Samorodnitsky's inequality to product probability distributions
• Determine optimal parameter λ for real q values and distributions
• Extend error-correcting code applications to finite alphabet codes

Innovation

• Generalized Samorodnitsky inequality to product distributions
• Extended optimal parameter range to real q values
• Applied generalized inequality to finite alphabet codes