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Let $T_{\epsilon}$ be the noise operator acting on functions on the booleancube $\{0,1\}^n$. 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 definedfunction of $q$.  We describe some applications for error-correcting codes and for matroids. Inparticular, we derive an upper bound on the weight distribution of duals ofBEC-capacity achieving binary linear codes. This improves the known bounds onthe linear-weight components of the weight distribution of constant rate binaryReed-Muller codes for almost all rates.

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