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Revision #1 to TR18-168 | 12th November 2019 09:06

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

Revision #1
Authors: Alex Samorodnitsky
Accepted on: 12th November 2019 09:06
Keywords:

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.

Changes to previous version:

A new version with some improved bounds.

### Paper:

TR18-168 | 25th September 2018 18:00

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

TR18-168
Authors: Alex Samorodnitsky
Publication: 7th October 2018 16:43
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 $q$.