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

TL;DR

This paper analyzes the soundness threshold of the Cube versus Cube test, establishing near-optimal bounds that improve previous results and have implications for low degree testing in finite fields.

Contribution

It provides a near-optimal analysis of the Cube versus Cube test, showing its soundness is polynomial in degree over the field size, improving prior bounds.

Findings

Soundness is polynomial in degree over field size, i.e., poly(d)/q.

Achieves optimal dependence on q for low degree testing.

Improves previous soundness bounds of poly(d)/q^{1/2}.

Abstract

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 $\mathbb{F}_q^n$, 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 $C_1\cap C_2$ is at least some $\epsilon$. An element of interest is the 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 ${\sf poly}(d)/q$. This result achieves the optimal dependence on $q$ for soundness in low degree testing and improves upon previous soundness results of ${\sf poly}(d)/q^{1/2}$ due to Bhangale, Dinur and Navon.