Breaking the ε-Soundness Bound of the Linearity Test over GF (2)

T. Kaufman, Simon Litsyn, Ning Xie

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

For Boolean functions that are ε-away from the set of linear functions, we study the lower bound on the rejection probability (denoted by rej(ε)) of the linearity test suggested by Blum, Luby and Rubinfeld. This problem is one of the most extensively studied problems in property testing of Boolean functions. The previously best bounds for rej(ε) were obtained by Bellare, Coppersmith, Håstad, Kiwi and Sudan. They used Fourier analysis to show that rej(ε) https://static-content.springer.com/image/chp%3A10.1007%2F978-3-540-85363-3_39/MediaObjects/978-3-540-85363-3_39_IEq1_HTML.png for every 0≤ϵ≤120≤ϵ≤12. They also conjectured that this bound might not be tight for ε's that are close to 1/2. In this paper we show that this indeed is the case. Specifically, we improve the lower bound of rej(ε) ≥ ε by an additive term that depends only on ε: rej(ε) ≥ϵ+min{1376ϵ3(1−2ϵ)12,14ϵ(1−2ϵ)4}≥ϵ+min{1376ϵ3(1−2ϵ)12,14ϵ(1−2ϵ)4}, for every 0≤ϵ≤120≤ϵ≤12. Our analysis is based on a relationship between rej(ε) and the weight distribution of a coset of the Hadamard code. We use both Fourier analysis and coding theory tools to estimate this weight distribution.
Original languageAmerican English
Title of host publicationApproximation, Randomization and Combinatorial Optimization. Algorithms and Techniques
EditorsAshish Goel, Klaus Jansen, José D. P. Rolim, Ronitt Rubinfeld
PublisherSpringer Berlin Heidelberg
StatePublished - 2008

Bibliographical note

Place of conference:USA

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