Approaching the Soundness Barrier: A Near Optimal Analysis of the Cube versus Cube Test
share
The Cube versus Cube test is a variant of the well-known Plane versus Plane test of Raz and Safra, in which to each 3-dimensional affine subspace C of 𝔽_q^n, a polynomial of degree at most d, T(C), is assigned in a somewhat locally consistent manner: taking two cubes C_1, C_2 that intersect in a plane uniformly at random, the probability that T(C_1) and T(C_2) agree on C_1∩ C_2 is at least some ϵ. An element of interest is the soundness threshold of this test, i.e. the smallest value of ϵ, such that this amount of local consistency implies a global structure; namely, that there is a global degree d function g such that g|_C≡ T(C) for at least Ω(ϵ) fraction of the cubes. We show that the cube versus cube low degree test has soundness poly(d)/q. This result achieves the optimal dependence on q for soundness in low degree testing and improves upon previous soundness results of poly(d)/q^1/2 due to Bhangale, Dinur and Navon.
READ FULL TEXT