Sub-Constant Error Probabilistically Checkable Proof of Almost-Linear Size

Abstract

We show a construction of a PCP with both sub-constant error and almost-linear size. Specifically, for some constant 0 < α < 1, we construct a PCP verifier for checking satisfiability of Boolean formulas that on input of size n uses $$\log\, n+O((\log\, n)^{1-\alpha})$$ random bits to make 7 queries to a proof of size $$n·2^{O((\log\, n)^{1-\alpha})}$$ , where each query is answered by $$O((\log\, n)^{1-\alpha})$$ bit long string, and the verifier has perfect completeness and error $$2^{-\Omega((\log\, n)^{\alpha})}$$ . The construction is by a new randomness-efficient version of the aggregation through curves technique. Its main ingredients are a recent low degree test with both sub-constant error and almost-linear size and a new method for constructing a short list of balanced curves.