An Upper Bound on ℓq Norms of Noisy Functions

Let Tϵ, 0 ≤ ϵ ≤ 1/2, be the noise operator acting on functions on the boolean cube 0,1\n. Let f be a nonnegative function on 0,1n and let q ≥ 1. We upper bound the ℓ q norm of Tϵ f by the average ℓ q norm of conditional expectations of f, given sets of roughly 1-2 ϵ rq 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.