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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CCC.2022.31
URN: urn:nbn:de:0030-drops-165934
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16593/
Bhandari, Siddharth ;
Harsha, Prahladh ;
Saptharishi, Ramprasad ;
Srinivasan, Srikanth
Vanishing Spaces of Random Sets and Applications to Reed-Muller Codes
Abstract
We study the following natural question on random sets of points in ?₂^m:
Given a random set of k points Z = {z₁, z₂, … , z_k} ⊆ ?₂^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},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16593},
URN = {urn:nbn:de:0030-drops-165934},
doi = {10.4230/LIPIcs.CCC.2022.31},
annote = {Keywords: Reed-Muller codes, polynomials, weight-distribution, vanishing ideals, erasures, capacity}
}
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Keywords: |
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Reed-Muller codes, polynomials, weight-distribution, vanishing ideals, erasures, capacity |
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Collection: |
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37th Computational Complexity Conference (CCC 2022) |
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Issue Date: |
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2022 |
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Date of publication: |
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11.07.2022 |
