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.