Short locally testable codes and proofs: a survey in two parts
Property testing
Short locally testable codes and proofs: a survey in two parts
Property testing
Short locally testable codes and proofs
Studies in complexity and cryptography
SIAM Journal on Discrete Mathematics
Applying cube attacks to stream ciphers in realistic scenarios
Cryptography and Communications
High dimensional expanders and property testing
Proceedings of the 5th conference on Innovations in theoretical computer science
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For Boolean functions that are $\epsilon$-far from the set of linear functions, we study the lower bound on the rejection probability (denoted by $\textsc{rej}(\epsilon)$) 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 $\textsc{rej}(\epsilon)$ were obtained by Bellare et al. [IEEE Trans. Inform. Theory, 42 (1996), pp. 1781-1795]. They used Fourier analysis to show that $\textsc{rej}(\epsilon)\geq\epsilon$ for every $0\leq\epsilon\leq1/2$. 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 $0\leq\epsilon\leq1/2$. Our analysis is based on a relationship between $\textsc{rej}(\epsilon)$ 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.