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Vanishing Spaces of Random Sets and Applications to Reed-Muller Codes

Authors Siddharth Bhandari , Prahladh Harsha , Ramprasad Saptharishi , Srikanth Srinivasan

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Siddharth Bhandari
  • Simons Institute for the Theory of Computing, Berkeley, CA, USA
Prahladh Harsha
  • Tata Institute of Fundamental Research, Mumbai, India
Ramprasad Saptharishi
  • Tata Institute of Fundamental Research, Mumbai, India
Srikanth Srinivasan
  • Aarhus University, Denmark


We thank the anonymous referees for several helpful comments.

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Siddharth Bhandari, Prahladh Harsha, Ramprasad Saptharishi, and Srikanth Srinivasan. Vanishing Spaces of Random Sets and Applications to Reed-Muller Codes. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 31:1-31:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


We study the following natural question on random sets of points in 𝔽₂^m: Given a random set of k points Z = {z₁, z₂, … , z_k} ⊆ 𝔽₂^m, what is the dimension of the space of degree at most r multilinear polynomials that vanish on all points in Z? We show that, for r ≤ γ m (where γ > 0 is a small, absolute constant) and k = (1-ε)⋅binom(m, ≤ r) for any constant ε > 0, the space of degree at most r multilinear polynomials vanishing on a random set Z = {z_1,…, z_k} has dimension exactly binom(m, ≤ r) - k with probability 1 - o(1). This bound shows that random sets have a much smaller space of degree at most r multilinear polynomials vanishing on them, compared to the worst-case bound (due to Wei (IEEE Trans. Inform. Theory, 1991)) of binom(m, ≤ r) - binom(log₂ k, ≤ r) ≫ binom(m, ≤ r) - k. Using this bound, we show that high-degree Reed-Muller codes (RM(m,d) with d > (1-γ) m) "achieve capacity" under the Binary Erasure Channel in the sense that, for any ε > 0, we can recover from (1-ε)⋅binom(m, ≤ m-d-1) random erasures with probability 1 - o(1). This also implies that RM(m,d) is also efficiently decodable from ≈ binom(m, ≤ m-(d/2)) random errors for the same range of parameters.

Subject Classification

ACM Subject Classification
  • Theory of computation → Error-correcting codes
  • Reed-Muller codes
  • polynomials
  • weight-distribution
  • vanishing ideals
  • erasures
  • capacity


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