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Low-soundness direct-product testers and PCPs from Kaufman--Oppenheim complexes

Ryan O'Donnell, Noah G. Singer

TL;DR

The paper delivers an elementary, strongly explicit sparse two-query direct-product tester based on Kaufman-Oppenheim coset complexes, answering open questions about low-soundness testing and enabling PCPs with arbitrarily small constant soundness and quasilinear length. The core technical advance is a dimension-independent $1$-coboundary expansion result for selected pinned subcomplexes, proved by an induction/restriction strategy on KO links and a Dehn-method-guided presentation bound for matrix-unipotent groups over $F_p[t]$. By combining high-dimensional expansion techniques with a careful group-theoretic treatment of unipotent matrices, the work achieves a concrete testing construction that avoids deep number theory in the PCP synthesis and demonstrates broad applicability to sparse direct-product testing. The results thus advance explicit combinatorial constructions of PCPs and shed light on the algebraic underpinnings required to realize robust, low-soundness testing via elementary, strongly explicit complexes.

Abstract

We study the Kaufman--Oppenheim coset complexes (STOC 2018, Eur. J. Comb. 2023), which have an elementary and strongly explicit description. Answering an open question of Kaufman, Oppenheim, and Weinberger (STOC 2025), we show that they support sparse direct-product testers in the low soundness regime. Our proof relies on the HDX characterization of agreement testing by Bafna--Minzer and Dikstein--Dinur (both STOC 2024), the recent result of Kaufman. et al, and follows techniques from Bafna--Lifshitz--Minzer and Dikstein--Dinur--Lubotzky (both FOCS 2024). Ultimately, the task reduces to showing dimension-independent coboundary expansion of certain $2$-dimensional subcomplexes of the KO complex; following the ``Dehn method'' of Kaufman and Oppenheim (ICALP 2021), we do this by establishing efficient presentation bounds for certain matrix groups over polynomial rings. As shown by Bafna, Minzer, and Vyas (STOC 2025), a consequence of our direct-product testing result is that the Kaufman--Oppenheim complexes can also be used to obtain PCPs with arbitrarily small constant soundness and quasilinear length. Thus the use of sophisticated number theory and algebraic group-theoretic tools in the construction of these PCPs can be avoided.

Low-soundness direct-product testers and PCPs from Kaufman--Oppenheim complexes

TL;DR

The paper delivers an elementary, strongly explicit sparse two-query direct-product tester based on Kaufman-Oppenheim coset complexes, answering open questions about low-soundness testing and enabling PCPs with arbitrarily small constant soundness and quasilinear length. The core technical advance is a dimension-independent -coboundary expansion result for selected pinned subcomplexes, proved by an induction/restriction strategy on KO links and a Dehn-method-guided presentation bound for matrix-unipotent groups over . By combining high-dimensional expansion techniques with a careful group-theoretic treatment of unipotent matrices, the work achieves a concrete testing construction that avoids deep number theory in the PCP synthesis and demonstrates broad applicability to sparse direct-product testing. The results thus advance explicit combinatorial constructions of PCPs and shed light on the algebraic underpinnings required to realize robust, low-soundness testing via elementary, strongly explicit complexes.

Abstract

We study the Kaufman--Oppenheim coset complexes (STOC 2018, Eur. J. Comb. 2023), which have an elementary and strongly explicit description. Answering an open question of Kaufman, Oppenheim, and Weinberger (STOC 2025), we show that they support sparse direct-product testers in the low soundness regime. Our proof relies on the HDX characterization of agreement testing by Bafna--Minzer and Dikstein--Dinur (both STOC 2024), the recent result of Kaufman. et al, and follows techniques from Bafna--Lifshitz--Minzer and Dikstein--Dinur--Lubotzky (both FOCS 2024). Ultimately, the task reduces to showing dimension-independent coboundary expansion of certain -dimensional subcomplexes of the KO complex; following the ``Dehn method'' of Kaufman and Oppenheim (ICALP 2021), we do this by establishing efficient presentation bounds for certain matrix groups over polynomial rings. As shown by Bafna, Minzer, and Vyas (STOC 2025), a consequence of our direct-product testing result is that the Kaufman--Oppenheim complexes can also be used to obtain PCPs with arbitrarily small constant soundness and quasilinear length. Thus the use of sophisticated number theory and algebraic group-theoretic tools in the construction of these PCPs can be avoided.

Paper Structure

This paper contains 98 sections, 126 theorems, 256 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. High-level description of components
  3. Direct-product testing
  4. An elementary, sparse, low-soundness agreement tester
  5. The local KO complex.
  6. Triword expansion
  7. Proofs of identity in the unipotent group
  8. Circuits
  9. "Relations" among circuits
  10. Avoidant transformations
  11. The power of working space
  12. The Steinberg relations
  13. Quantitative aspects of the proof
  14. Preliminaries
  15. Indexed simplicial complexes
  16. ...and 83 more sections

Key Result

theorem 1.1

Given $\delta > 0$, provided $k \geq \exp(\operatorname{poly}(1/\delta))$, $n \geq \exp(\operatorname{poly}(k))$, and $p \geq \operatorname{poly}(n)$ is prime, the KOW25 complexes $\{ \mathfrak{A}_n(\mathbb F_{Q}) : Q = p^s,\ s > 3n\}$ yield a strongly explicit family of $k$-uniform hypergraphs with

Figures (8)

  • Figure 1: A schematic diagram of our proof; the contributions of our work are shaded. Rounded rectangles are "reduction" theorems.
  • Figure 2: Some circuits. The circuits in \ref{['fig:steinberg:a', 'fig:steinberg:b']} are equivalent (the "linearity" relation, \ref{['ex:steinberg:linear']}). The circuits in \ref{['fig:steinberg:c', 'fig:steinberg:d']} are equivalent (the "nontrivial commutator" relation, \ref{['ex:steinberg:nontrivial']}). The circuit in \ref{['fig:steinberg:e']} is equivalent to the trivial (empty) circuit (the "trivial commutator" relation, \ref{['ex:steinberg:trivial']}).
  • Figure 3: An illustration of the situation when $n = 4$. There are $n+1=5$ wires, labeled $0,\ldots,4$, and $n$ moats, labeled $1, \ldots,4$. The "gate" from wire $1$ to wire $3$ crosses moats $2$ and $3$ but not moats $1$ or $4$. This corresponds to the fact that $(1\mathinner{.\,.}3] = \{2,3\}$.
  • Figure 4: The circuit in \ref{['ex:bd']} with $n=3$, $a=1$, $b=2$, and $c=3$. Every gate avoids at least one moat; in particular, the $\pm p$ gates avoid $c$ and the $\pm q$ gates avoid $a$.
  • Figure 5: The utility of additional working space.
  • ...and 3 more figures

Theorems & Definitions (341)

  • theorem 1.1: Direct-product testing application theorem
  • theorem 1.2: KO link coboundary expansion
  • theorem 1.3: Separated restrictions have dimension-independent coboundary expansion
  • theorem 1.3: Unipotent group presentation bounds
  • definition 2.1: Agreement tester soundness
  • definition 2.8: Trivial $1$-cohomology over $\Gamma$
  • theorem 2.9
  • definition 2.10: $1$-coboundary expansion
  • example 2.11: "Linearity" relation
  • example 2.12: "Nontrivial commutator" relation
  • ...and 331 more