PCPs via low-degree long code and hardness for constrained hypergraph coloring

We develop new techniques to incorporate the recently proposed "short code" (a low-degree version of the long code) into the construction and analysis of PCPs in the classical "Label Cover + Fourier Analysis" framework. As a result, we obtain more size-efficient PCPs that yield improved hardness results for approximating CSPs and certain coloringtype problems. In particular, we show a hardness for a variant of hypergraph coloring (with hyperedges of size 6), with a gap between 2 and exp(2Ω( √ log logN)) number of colors where N is the number of vertices. This is the first hardness result to go beyond the O(logN) barrier for a coloring-type problem. Our hardness bound is a doubly exponential improvement over the previously known O(log logN)-coloring hardness for 2- colorable hypergraphs, and an exponential improvement over the (logN)Ω(1)-coloring hardness for O(1)-colorable hypergraphs. Stated in terms of "covering complexity," we show that for 6-ary Boolean CSPs, it is hard to decide if a given instance is perfectly satisfiable or if it requires more than 2Ω( √ log logN) assignments for covering all of the constraints. While our methods do not yield a result for conventional hypergraph coloring due to some technical reasons, we also prove hardness of (logN)Ω(1)-coloring 2-colorable 6-uniform hypergraphs (this result relies just on the long code). A key algebraic result driving our analysis concerns a very low-soundness error testing method for Reed-Muller codes. We prove that if a function β : F^m₂ → F₂ is 2Ω(d) far in absolute distance from polynomials of degree m-d, then the probability that deg(βg) ≤ m-3d/4 for a random degree d/4 polynomial g is doubly exponentially small in d.