Optimal constant in Plane vs Plane low-degree test soundness

Determine the largest constant c for which the Plane vs Plane low-degree test over finite fields has soundness at probability q^{-c}: specifically, prove tight bounds on c such that if a table assigning degree-d functions to planes in F_q^n passes the Plane vs Plane test with probability at least q^{-c}, then there exists a global degree-d polynomial f whose restrictions agree with the table on a positive (quantified) fraction of planes. This problem is directly tied to optimizing the soundness–alphabet–instance size tradeoff in PCPs.

Background

The Plane vs Plane test encodes a low-degree function by its restrictions to 2-dimensional affine subspaces and tests consistency on intersections. While perfect completeness is straightforward, quantitative soundness parameters are crucial for PCP applications. Existing analyses establish soundness only for specific ranges of the test passing probability, with the best known constant c for the threshold being 1/8 for the Plane vs Plane test.

The authors highlight that optimizing c affects the achievable alphabet-size vs soundness vs instance-size tradeoff in PCP constructions. Better bounds for c directly enable stronger inapproximability results and more efficient PCP parameters.