An upper bound on $\ell_q$ norms of noisy functions

TL;DR

This paper establishes an upper bound on the $\, ext{ell}_q$ norms of noisy functions on the boolean cube, with applications to error-correcting codes and matroids, improving bounds on weight distributions of certain codes.

Contribution

It introduces a new upper bound on the $\, ext{ell}_q$ norm of noisy functions, linking it to conditional expectations over subsets of variables, with applications to coding theory.

Findings

Upper bound on $\, ext{ell}_q$ norms of noisy functions.

Improved bounds on weight distributions of Reed-Muller codes.

Applications to error-correcting codes and matroids.

Abstract

Let $T_{\epsilon}$ 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 $q \ge 1$. We upper bound the $\ell_q$ norm of $T_{\epsilon} f$ by the average $\ell_q$ norm of conditional expectations of $f$, given sets of roughly $(1-2\epsilon)^{r(q)} \cdot n$ variables, where $r$ is an explicitly defined function of $q$. We describe some applications for error-correcting codes and for matroids. In particular, we derive an upper bound on the weight distribution of duals of BEC-capacity achieving binary linear codes. This improves the known bounds on the linear-weight components of the weight distribution of constant rate binary Reed-Muller codes for almost all rates.