What are the best known asymptotic upper bounds on sizes of probabilistically checkable proofs? Ideally, I am looking for a contemporary survey on this broad question, but if there is none, I am especially interested in inapproximability of 3-SAT.
Let 7/8+ε-3-SAT be 3-SAT with the promise that if 7/8+ε fraction of the clauses are satisfiable, then the instance is satisfiable. What are the best known reductions of 3-SAT with $n$ clauses to 7/8+ε-3-SAT? For example, is there a reduction using $O(n \log n)$ clauses? ($O(n)$ clauses is an open problem.) A reduction in uniform quasilinear size NC? What is the dependence on $ε$, including when $ε→0$? Is there a known linear size (dependent on $ε$) reduction of (1-ε)-3-SAT to 7/8+ε-3-SAT, and if not, do we have better bounds for (1-ε)-3-SAT? Even a partial answer would be interesting.
Also, while it would perhaps make the question too broad, I should mention that another important issue here are the constant factors, which due to techniques like the long code are commonly infeasibly large.