## Abstract

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{1376ε^{3}(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.

Original language | English |
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Pages (from-to) | 1988-2003 |

Number of pages | 16 |

Journal | SIAM Journal on Computing |

Volume | 39 |

Issue number | 5 |

DOIs | |

State | Published - 2010 |

## Keywords

- Coding theory
- Fourier analysis
- Linearity test
- Property testing