Breaking the epsilon-Soundness Bound of the Linearity Test over Gf(2)

Abstract

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