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

TL;DR

This paper demonstrates that Kaufman--Oppenheim complexes support sparse direct-product testers with low soundness, enabling the construction of PCPs with arbitrarily small soundness and quasilinear length without complex number-theoretic tools.

Contribution

It establishes low-soundness direct-product testers on Kaufman--Oppenheim complexes and derives PCPs with small soundness and quasilinear length using these complexes.

Findings

Support for sparse direct-product testers in low soundness regime

Construction of PCPs with arbitrarily small constant soundness

Avoidance of complex number-theoretic tools in PCP construction

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.