Constant Degree Direct Product Testers with Small Soundness

TL;DR

This work resolves a conjecture of Dinur and Kaufman by constructing high‑dimensional expanders with bounded degree that admit 2‑query direct product testers with small soundness for all fixed $\delta>0$. The authors develop a general, non‑abelian coboundary expansion framework and prove a local‑to‑global theorem that lifts coboundary expansion in links to cosystolic expansion of the global complex. The technical novelty lies in a robust inductive approach and triangulation strategy for tripartite and symplectic Grassmannian type structures, enabling domination of coboundary constants by poly(r) factors with subexponential bounds. The Chapman–Lubotzky type complexes are shown to satisfy the necessary conditions, yielding explicit, efficiently computable LDX with constant degree that support strong direct product tests. This advances hardness amplification and PCP‑style constructions by providing sparse, high‑dimensional objects with provably good 2‑query product testers.

Abstract

Let $X$ be a $d$-dimensional simplicial complex. A function $F\colon X(k)\to \{0,1\}^k$ is said to be a direct product function if there exists a function $f\colon X(1)\to \{0,1\}$ such that $F(σ) = (f(σ_1), \ldots, f(σ_k))$ for each $k$-face $σ$. In an effort to simplify components of the PCP theorem, Goldreich and Safra introduced the problem of direct product testing, which asks whether one can test if $F\colon X(k)\to \{0,1\}^k$ is correlated with a direct product function by querying $F$ on only $2$ inputs. Dinur and Kaufman 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 $δ>0$, there exists a family of high-dimensional expanders with degree $O_δ(1)$ and a $2$-query direct product tester with soundness $δ$. We use the characterization given by a subset of the authors and independently by Dikstein and Dinur, 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 Chapman and Lubotzky satisfies the necessary conditions, thus admitting a 2-query direct product tester with small soundness.

Theorems & Definitions (101)

• Definition 1.1
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• Theorem 1.3
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• Remark 1.10