Sub-Constant Error Probabilistically Checkable Proof of Almost-Linear Size
We show a construction of a PCP with both sub- constant error and almost- linear size. Specifically, for some constant 0< alpha< 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.
sub-constant error almost-linear size balanced curves probabilistically checkable proof (pcp)