A Query Efficient Non-adaptive Long Code Test with Perfect Completeness

Suguru Tamaki, Yuichi Yoshida

Abstract: Long Code testing is a fundamental problem in the area of property testing and hardness of approximation. Long Code is a function of the form f(x) = x i for some index i. In the Long Code testing, the problem is, given oracle access to a collection of Boolean functions, to decide whether all the functions are the same Long Code, or cross-influences of any two functions are small. In this paper, we study the following problem: How small the soundness s of the Long Code test with perfect completeness can be by using non-adaptive q queries? We give a Long Code test with s = (2q + 3)/2q, where q is of the form 2k − 1 for any integer k > 2. Our test is a “noisy” version of Samorodnitsky-Trevisan’s Hyper Graph linearity test with suitably chosen noise distribution. To bound the soundness, we use Invariance-Principle style analysis in the spirit of O’Donnell and Wu (STOC 2009).