Breaking the e-soundness bound of the linearity test over gf(2)

For Boolean functions that are e-far from the set of linear functions, we study the lower bound on the rejection probability (denoted by REJ(ε)) 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(ε) were obtained by Bellare et al. [IEEE Trans. Inform. Theory,42 (1996), pp. 1781-1795]. They used Fourier analysis to show that REJ(ε) ≥ e for every 0 ≤ ε ≤ 1/2. They also conjectured that this bound might not be tight for e'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(ε) ≥ e by an additive constant that depends only on e: REJ(ε) ≥ ε + min{1376ε³(l - 2ε) ¹², 1/4ε(l - 2ε)⁴}, for every 0 ≤ ε ≤ 1 /2. Our analysis is based on a relationship between REJ(ε) 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.