TY - GEN
T1 - Breaking the ε-soundness bound of the linearity test over GF(2)
AU - Kaufman, Tali
AU - Litsyn, Simon
AU - Xie, Ning
PY - 2008
Y1 - 2008
N2 - For Boolean functions that are ε-away 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. This problem is one of the most extensively studied problems in property testing of Boolean functions. The previously best bounds for rej(ε) were obtained by Bellare, Coppersmith, Håstad, Kiwi and Sudan. They used Fourier analysis to show that REJ(ε) ≥ ε for every 0 ≤ ε ≤ 1/2. They also conjectured that this bound might not be tight for ε'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(ε) ≥ ε by an additive term that depends only on ε: REJ(ε)≥ ε+min {1376ε3(1 - 2ε)12, 1/4 ε (1 - 2 ε)4}, for every 0 ≤ ε ≤ 1/2. Our analysis is based on a relationship between REJ(ε) 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.
AB - For Boolean functions that are ε-away 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. This problem is one of the most extensively studied problems in property testing of Boolean functions. The previously best bounds for rej(ε) were obtained by Bellare, Coppersmith, Håstad, Kiwi and Sudan. They used Fourier analysis to show that REJ(ε) ≥ ε for every 0 ≤ ε ≤ 1/2. They also conjectured that this bound might not be tight for ε'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(ε) ≥ ε by an additive term that depends only on ε: REJ(ε)≥ ε+min {1376ε3(1 - 2ε)12, 1/4 ε (1 - 2 ε)4}, for every 0 ≤ ε ≤ 1/2. Our analysis is based on a relationship between REJ(ε) 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.
UR - https://www.scopus.com/pages/publications/51849129240
U2 - 10.1007/978-3-540-85363-3_39
DO - 10.1007/978-3-540-85363-3_39
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AN - SCOPUS:51849129240
SN - 3540853626
SN - 9783540853626
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 498
EP - 511
BT - Approximation, Randomization and Combinatorial Optimization
T2 - 11th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2008 and 12th International Workshop on Randomization and Computation, RANDOM 2008
Y2 - 25 August 2008 through 27 August 2008
ER -