{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article2197","name":"Breaking the $\\epsilon$-Soundness Bound of the Linearity Test over GF(2)","abstract":"For Boolean functions that are $epsilon$-far from the set of linear functions, \r\nwe study the lower bound on the rejection probability (denoted by $extsc{rej}(epsilon)$) of the linearity test suggested by Blum, Luby and Rubinfeld. \r\nThis problem is arguably the most fundamental and extensively studied problem in property testing of Boolean functions. \r\n\r\nThe previously best bounds for $extsc{rej}(epsilon)$ were obtained by Bellare,\r\nCoppersmith, H{{a}}stad, Kiwi and Sudan. They used Fourier analysis\r\nto show that $\textsc{rej}(epsilon) geq e$ for every $0 leq epsilon leq\r\nfrac{1}{2}$. They also conjectured that this bound might not be tight for\r\n$epsilon$'s which are close to $1\/2$. In this paper we show that this indeed is\r\nthe case. Specifically, we improve the lower bound of $\textsc{rej}(epsilon) geq\r\nepsilon$ by an additive constant that depends only on $epsilon$:\r\n$extsc{rej}(epsilon) geq epsilon + min {1376epsilon^{3}(1-2epsilon)^{12},\r\nfrac{1}{4}epsilon(1-2epsilon)^{4}}$, for every $0 leq epsilon leq frac{1}{2}$.\r\nOur analysis is based on a relationship between $extsc{rej}(epsilon)$ and the\r\nweight distribution of a coset of the Hadamard code. We use both Fourier\r\nanalysis and coding theory tools to estimate this weight distribution.","keywords":["Linearity test","Fourier analysis","coding theory"],"author":[{"@type":"Person","name":"Kaufman, Tali","givenName":"Tali","familyName":"Kaufman"},{"@type":"Person","name":"Litsyn, Simon","givenName":"Simon","familyName":"Litsyn"},{"@type":"Person","name":"Xie, Ning","givenName":"Ning","familyName":"Xie"}],"position":3,"pageStart":1,"pageEnd":0,"dateCreated":"2008-11-25","datePublished":"2008-11-25","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Kaufman, Tali","givenName":"Tali","familyName":"Kaufman"},{"@type":"Person","name":"Litsyn, Simon","givenName":"Simon","familyName":"Litsyn"},{"@type":"Person","name":"Xie, Ning","givenName":"Ning","familyName":"Xie"}],"copyrightYear":"2008","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.08341.3","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume712","volumeNumber":8341,"name":"Dagstuhl Seminar Proceedings, Volume 8341","dateCreated":"2008-11-25","datePublished":"2008-11-25","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article2197","isPartOf":{"@type":"Periodical","@id":"#series119","name":"Dagstuhl Seminar Proceedings","issn":"1862-4405","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume712"}}}