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.