We show a construction of a PCP with both sub-constant error and
almost-linear size. Specifically, for some constant alpha in (0,1),
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 query a constant number of places
in a proof of size n 2^{O((log n)^{1-alpha})} over
symbols consisting of O((log n)^{1-alpha}) bits and achieves
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.