When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CCC.2022.31
URN: urn:nbn:de:0030-drops-165934
URL: https://drops.dagstuhl.de/opus/volltexte/2022/16593/
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### Vanishing Spaces of Random Sets and Applications to Reed-Muller Codes

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### Abstract

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.

### BibTeX - Entry

```@InProceedings{bhandari_et_al:LIPIcs.CCC.2022.31,
author =	{Bhandari, Siddharth and Harsha, Prahladh and Saptharishi, Ramprasad and Srinivasan, Srikanth},
title =	{{Vanishing Spaces of Random Sets and Applications to Reed-Muller Codes}},
booktitle =	{37th Computational Complexity Conference (CCC 2022)},
pages =	{31:1--31:14},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-241-9},
ISSN =	{1868-8969},
year =	{2022},
volume =	{234},
editor =	{Lovett, Shachar},
publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},