eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-07-11
31:1
31:14
10.4230/LIPIcs.CCC.2022.31
article
Vanishing Spaces of Random Sets and Applications to Reed-Muller Codes
Bhandari, Siddharth
1
https://orcid.org/0000-0003-3481-6078
Harsha, Prahladh
2
https://orcid.org/0000-0002-2739-5642
Saptharishi, Ramprasad
2
https://orcid.org/0000-0002-7485-3220
Srinivasan, Srikanth
3
https://orcid.org/0000-0001-6491-124X
Simons Institute for the Theory of Computing, Berkeley, CA, USA
Tata Institute of Fundamental Research, Mumbai, India
Aarhus University, Denmark
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol234-ccc2022/LIPIcs.CCC.2022.31/LIPIcs.CCC.2022.31.pdf
Reed-Muller codes
polynomials
weight-distribution
vanishing ideals
erasures
capacity