Breaking the ε-Soundness Bound of the Linearity Test over GF(2)

• Authors: Tali Kaufman;Simon Litsyn;Ning Xie
• Venue: APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
• Year: 2008

For Boolean functions that are 茂戮驴-away from the set of linear functions, we study the lower bound on the rejection probability (denoted by rej(茂戮驴)) of the linearity test suggested by Blum, Luby and Rubinfeld. This problem is one of the most extensively studied problems in property testing of Boolean functions.The previously best bounds for rej(茂戮驴) were obtained by Bellare, Coppersmith, Håstad, Kiwi and Sudan. They used Fourier analysis to show that rej(茂戮驴) for every $0 \leq \epsilon \leq \frac{1}{2}$. They also conjectured that this bound might not be tight for 茂戮驴's that are close to 1/2. In this paper we show that this indeed is the case. Specifically, we improve the lower bound of rej(茂戮驴) 茂戮驴 茂戮驴by an additive term that depends only on 茂戮驴: rej(茂戮驴) $\geq \epsilon + \min \{1376\epsilon^{3}(1-2\epsilon)^{12}, \frac{1}{4}\epsilon(1-2\epsilon)^{4}\}$, for every $0 \leq \epsilon \leq \frac{1}{2}$. Our analysis is based on a relationship between rej(茂戮驴) 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.