TY - JOUR

T1 - Breaking the ϵ-soundness bound of the linearity test over GF (2)

AU - Kaufman, T.

AU - Litsyn, Simon

AU - Xie, Ning

PY - 2010

Y1 - 2010

N2 - For Boolean functions that are $\epsilon$-far from the set of linear functions, we study the lower bound on the rejection probability (denoted by $\textsc{rej}(\epsilon)$) 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 $\textsc{rej}(\epsilon)$ were obtained by Bellare et al. [IEEE Trans. Inform. Theory, 42 (1996), pp. 1781–1795]. They used Fourier analysis to show that $\textsc{rej}(\epsilon)\geq\epsilon$ for every $0\leq\epsilon\leq1/2$. 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 $\textsc{rej}(\epsilon)\geq\epsilon$ by an additive constant that depends only on $\epsilon$: $\textsc{rej}(\epsilon)\geq\epsilon+\min\{1376\epsilon^{3}(1-2\epsilon)^{12},\frac{1}{4}\epsilon(1-2\epsilon)^{4}\}$, for every $0\leq\epsilon\leq1/2$. Our analysis is based on a relationship between $\textsc{rej}(\epsilon)$ 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.
Read More: http://epubs.siam.org/doi/abs/10.1137/080715548

AB - For Boolean functions that are $\epsilon$-far from the set of linear functions, we study the lower bound on the rejection probability (denoted by $\textsc{rej}(\epsilon)$) 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 $\textsc{rej}(\epsilon)$ were obtained by Bellare et al. [IEEE Trans. Inform. Theory, 42 (1996), pp. 1781–1795]. They used Fourier analysis to show that $\textsc{rej}(\epsilon)\geq\epsilon$ for every $0\leq\epsilon\leq1/2$. 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 $\textsc{rej}(\epsilon)\geq\epsilon$ by an additive constant that depends only on $\epsilon$: $\textsc{rej}(\epsilon)\geq\epsilon+\min\{1376\epsilon^{3}(1-2\epsilon)^{12},\frac{1}{4}\epsilon(1-2\epsilon)^{4}\}$, for every $0\leq\epsilon\leq1/2$. Our analysis is based on a relationship between $\textsc{rej}(\epsilon)$ 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.
Read More: http://epubs.siam.org/doi/abs/10.1137/080715548

UR - https://scholar.google.co.il/scholar?q=Breaking+the+Epsilon-Soundness+Bound+of+the+Linearity+Test+over+GF+%2C+Tali+Kaufman&btnG=&hl=en&as_sdt=0%2C5

M3 - Article

SN - 0097-5397

VL - 39

SP - 1988

EP - 2003

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

IS - 5

ER -