2026-10-07T14:03:53Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:23704 2025-10-27T09:42:37Z ddc:004 open_access

Biased Linearity Testing in the 1% Regime Khot, Subhash Mittal, Kunal Linearity test 1% regime p-biased We study linearity testing over the p-biased hypercube ({0,1}ⁿ, μ_p^{⊗n}) in the 1% regime. For a distribution ν supported over {x ∈ {0,1}^k:∑_{i=1}^k x_i = 0 (mod 2)}, with marginal distribution μ_p in each coordinate, the corresponding k-query linearity test Lin(ν) proceeds as follows: Given query access to a function f:{0,1}ⁿ → {-1,1}, sample (x_1,… ,x_k)∼ ν^{⊗n}, query f on x_1,… ,x_k, and accept if and only if ∏_{i ∈ [k]} f(x_i) = 1. Building on the work of Bhangale, Khot, and Minzer (STOC '23), we show, for 0 < p ≤ 1/2, that if k ≥ 1+1/p, then there exists a distribution ν such that the test Lin(ν) works in the 1% regime; that is, any function f:{0,1}ⁿ → {-1,1} passing the test Lin(ν) with probability ≥ 1/2+ε, for some constant ε > 0, satisfies Pr_{x∼μ_p^{⊗n}}[f(x) = g(x)] ≥ 1/2+δ, for some linear function g, and a constant δ = δ(ε) > 0. Conversely, we show that if k < 1+1/p, then no such test Lin(ν) works in the 1% regime. Our key observation is that the linearity test Lin(ν) works if and only if the distribution ν satisfies a certain pairwise independence property. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Subhash Khot and Kunal Mittal 2025 Is Part Of LIPIcs, Volume 339, 40th Computational Complexity Conference (CCC 2025) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.CCC.2025.10 urn:nbn:de:0030-drops-237046 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2025.10 eng https://creativecommons.org/licenses/by/4.0/legalcode