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TR07-098 | 2nd October 2007 00:00

#### Breaking the $\epsilon$-Soundness Bound of the Linearity Test over GF(2)

TR07-098
Authors: Tali Kaufman, Simon Litsyn, Ning Xie
Publication: 13th October 2007 12:05
Keywords:

Abstract:

For Boolean functions that are $\epsilon$-far from the set of linear functions, we study the lower bound on the rejection probability (denoted $\textsc{rej}(\epsilon)$) of the linearity test suggested by Blum, Luby and Rubinfeld. The interest in this problem is partly due to its relation to PCP constructions and hardness of approximating some
NP-hard problems. It seems that the problem of lower bounding
$\textsc{rej}(\epsilon)$ becomes more difficult as $\epsilon$
approaches $1/2$.

The previously best bounds for $\textsc{rej}(\epsilon)$ were obtained
by Bellare, Coppersmith, H{\r{a}}stad, Kiwi and Sudan. They used Fourier analysis to show that $\textsc{rej}(\epsilon) \geq \e$ for every $0 \leq \epsilon \leq \frac{1}{2}$. 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 $\textsc{rej}(\epsilon) \geq \epsilon$ 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 $0 \leq \epsilon \leq \frac{1}{2}$. Our analysis is based on a relationship between
$\textsc{rej}(\epsilon)$ 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.

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