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

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) = xi 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). Previously, Hastad and Khot (Theory of Computing, 2005) had shown s = 24√q/2q for infinitely many q. Chen (RANDOM 2009) improved this to s = q3/2q for infinitely many q with "adaptive" queries. As for the Long Code test with "almost" perfect completeness, Samorodnitsky and Trevisan (SICOMP, 2009) have shown s = 2q/2q (or even (q + 1)/2q for infinitely many q). Austrin and Mossel (Computational Complexity, 2009) have improved this to s = (q+o(q))/2q (or even (q+4)/2q assuming the famous Hadamard Conjecture) for any q.

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