An Upper Bound on $\ell_q$ Norms of Noisy Functions

Abstract

Let ${T}_{ \epsilon }$ , $0 \le \epsilon \le 1/2$ , 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 BEC-capacity achieving binary linear codes and their duals. This improves the known bounds on the linear-weight components of the weight distribution of constant rate binary Reed-Muller codes for all (constant) rates.