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The Cube versus Cube test is a variant of the well-known Plane versus Planetest of Raz and Safra, in which to each $3$-dimensional affine subspace $C$ of$\mathbb{F}_q^n$, a polynomial of degree at most $d$, $T(C)$, is assigned in asomewhat locally consistent manner: taking two cubes $C_1, C_2$ that intersectin a plane uniformly at random, the probability that $T(C_1)$ and $T(C_2)$agree on $C_1\cap C_2$ is at least some $\epsilon$. An element of interest isthe soundness threshold of this test, i.e. the smallest value of $\epsilon$,such that this amount of local consistency implies a global structure; namely,that there is a global degree $d$ function $g$ such that $g|_{C} \equiv T(C)$for at least $\Omega(\epsilon)$ fraction of the cubes.  We show that the cube versus cube low degree test has soundness ${\sfpoly}(d)/q$. This result achieves the optimal dependence on $q$ for soundnessin low degree testing and improves upon previous soundness results of ${\sfpoly}(d)/q^{1/2}$ due to Bhangale, Dinur and Navon.

Approaching the Soundness Barrier: A Near Optimal Analysis of the Cube versus Cube Test