Breaking the epsilon-Soundness Bound of the Linearity Test over Gf(2)
Name
Kaufman-2010-BREAKING THE epsilon.pdf
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Author(s)
Kaufman-Halman, Tali
Litsyn, Simon
Xie, Ning
Alternative Title
BREAKING THE ε-SOUNDNESS BOUND OF THE LINEARITY TEST OVER GF(2)
Date Issued
February 2010
Journal
SIAM Journal of Computing
Publisher
Society for Industrial and Applied Mathematics
Citation
Kaufman, Tali, Simon Litsyn, and Ning Xie. "Breaking the epsilon-Soundness Bound of the Linearity Test over Gf(2)." SIAM Journal of Computing (2010) Volume 39, Issue 5 : pp. 1988-2003. ©2010 Society for Industrial and Applied Mathematics.
Version
Final published version
Abstract
For Boolean functions that are $\epsilon$-far from the set of linear functions, we study the lower bound on the rejection probability (denoted by REJ(epsilon) of the linearity test suggested by Blum, Luby, and Rubinfeld [J. Comput. System Sci., 47 (1993), pp. 549–595]. This problem is arguably the most fundamental and extensively studied problem in property testing of Boolean functions. The previously best bounds for REJ(epsilon) were obtained by Bellare et al. [IEEE Trans. Inform. Theory, 42 (1996), pp. 1781–1795]. They used Fourier analysis to show that REJ(epsilon)[geq]epsilon for every 0[leq]epsilon [leq]1/2. They also conjectured that this bound might not be tight for epsilon's which are close to 1/2. In this paper we show that this indeed is the case. Specifically, we improve the lower bound of REJ(epsilon)[geq]epsilon by an additive constant that depends only on epsilon: REJ(epsilon)[geq] epsilon+min{1376epsilon[superscript 3](1-2epsilon)[superscript 12],[frac 1 over 4epsilon(1-2epsilon[superscript 4]}, for every 0[leq]epsilon[leq]1/2. Our analysis is based on a relationship between REJ(epsilon) and the weight distribution of a coset code of the Hadamard code. We use both Fourier analysis and coding theory tools to estimate this weight distribution.
MIT Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1137/080715548
