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.
Bibliographical noteFunding Information:
Manuscript received November 19, 2018; accepted August 19, 2019. Date of publication September 30, 2019; date of current version January 20, 2020. This work was supported in part by ISF under Grant 1724/15. The author is with the School of Engineering and Computer Science, The Hebrew University of Jerusalem, Jerusalem 91904, Israel (e-mail: firstname.lastname@example.org). Communicated by E. Abbe, Associate Editor for Machine Learning. Color versions of one or more of the figures in this article are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIT.2019.2944698
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- Binary codes
- channel capacity