Parallel LTC amplification on HDX without soundness decay

Construct families of high dimensional expanders and locally testable codes that support a parallel amplification framework for local testability without asymptotic decay in soundness. Specifically, develop a q-step distribution generating k valid parallel tests and prove that the resulting parallel tester has no soundness loss compared to the base tester.

Background

The paper shows a parallel LTC amplification theorem on the complete complex with no soundness decay, but extending it to HDX would require matching the base tester’s queries via HDX walks and controlling correlations across parallel instances.

The authors formulate an explicit conjecture describing the desired HDX-and-code framework that would enable lossless parallel amplification, substantially improving LTC soundness on sparse structures.

References

Conjecture [Parallel LTC Amplification] There exists a family of high dimensional expanders and locally testable codes which admit a ``parallel'' variant of \pref{thm:LTC-ABNNR}, that is:

  1. There is a q-step random walk or distribution on $X$ generating $k$ valid `parallel' tests
  2. The resulting parallel tester has no (asymptotic) decay in soundness.
— Chernoff Bounds and Reverse Hypercontractivity on HDX (2404.10961 - Dikstein et al., 17 Apr 2024) in Section: Toward Lossless Amplification from HDX?, Conjecture [Parallel LTC Amplification]