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 < α < 1, we construct a PCP verifier for checking satisfia-bility of Boolean formulas that on input of size n uses logn+O((log n)1−α) random bits to query a constant number of places in a proof of size n · 2O((logn)1−α) over symbols consisting of O((log n)1−α) bits and achieves error 2−Ω((logn) α). The construction is by a new randomness-efficient version of the aggregation through curves technique [1]. Its main ingredients are a recent low degree test with both sub-constant error and almost-linear size [12] and a new method for constructing a short list of balanced curves