Constant Degree Direct Product Testers with Small Soundness

Let $X$ be a d-dimensional simplicial complex. A function $F: X(k)\rightarrow\{0,1\}^{k}$ is said to be a direct product function if there exists a function $f: x(1)\rightarrow\{0,1\}$ such that $F(\sigma)=(f(\sigma_{1}),\ \ldots,\ f(\sigma_{k}))$ for each k-face $\sigma$, In an effort to simplify components of the PCP theorem, Goldreich and Safra [1] introduced the problem of direct product testing, which asks whether one can test if $F: X(k)\rightarrow\{0,1\}^{k}$ - is correlated with a direct product function by querying $F$ on only 2 inputs. Dinur and Kaufman [2] conjectured that there exist bounded degree complexes with a direct product test in the small soundness regime. We resolve their conjecture by showing that for all $\delta > 0$, there exists a family of high-dimensional expanders with degree $O_{\delta}(1)$ and a 2-query direct product tester with soundness $\delta$ We use the characterization given by [3] and independently by [4], who showed that some form of non-Abelian coboundary expansion (which they called “Unique-Games coboundary expansion”) is a necessary and sufficient condition for a complex to admit such direct product testers. Our main technical contribution is a general technique for showing coboundary expansion of complexes with coefficients in a non-Abelian group. This allows us to prove that the high dimensional expanders constructed by [5] satisfy the conditions of [3], thus admitting a 2-query direct product tester with small soundness.