An Upper Bound on ℓq Norms of Noisy Functions

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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.

Original languageAmerican English
Article number8853269
Pages (from-to)742-748
Number of pages7
JournalIEEE Transactions on Information Theory
Issue number2
StatePublished - Feb 2020


  • Binary codes
  • channel capacity

All Science Journal Classification (ASJC) codes

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences


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