Achieving sub-polylogarithmic soundness in quasi-polynomial PCPs within the covering-property framework

Develop a PCP construction, ideally within the framework that composes a smooth parallel repetition outer PCP with a Grassmann-graph-based inner PCP relying on the covering property, that achieves soundness error smaller than inversely poly-logarithmic in the instance size N (e.g., better than 1/(log N)^C) while maintaining quasi-polynomial instance size and controlled alphabet size. This seeks progress toward soundness regimes relevant to the sliding scale direction but within the techniques analyzed in this work.

Background

The authors’ techniques yield nearly optimal alphabet-to-soundness tradeoffs but are limited to soundness errors that are inversely poly-logarithmic in the instance size. They explain that their approach—based on the covering property and particular composition—does not reach smaller soundness errors associated with the sliding scale direction.

Improving soundness below inverse poly-logarithmic while keeping quasi-polynomial size and reasonable alphabets would have significant implications for inapproximability results and might require overcoming structural barriers inherent in covering-property-based PCPs.