2026-07-26T01:16:05Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:16593 2024-03-06T10:57:53Z ddc:004 open_access

Vanishing Spaces of Random Sets and Applications to Reed-Muller Codes Bhandari, Siddharth Harsha, Prahladh Saptharishi, Ramprasad Srinivasan, Srikanth Reed-Muller codes polynomials weight-distribution vanishing ideals erasures capacity 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Siddharth Bhandari and Prahladh Harsha and Ramprasad Saptharishi and Srikanth Srinivasan 2022 Is Part Of LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.CCC.2022.31 urn:nbn:de:0030-drops-165934 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.31 eng https://creativecommons.org/licenses/by/4.0/legalcode