  We construct 2-query, quasi-linear sized probabilistically checkable proofs
(PCPs) with arbitrarily small constant soundness, improving upon Dinur's
2-query quasi-linear size PCPs with soundness $1-\Omega(1)$. As an immediate
corollary, we get that under the exponential time hypothesis, for all $\epsilon
>0$ no approximation algorithm for $3$-SAT can obtain an approximation ratio of
$7/8+\epsilon$ in time $2^{n/\log^C n}$, where $C$ is a constant depending on
$\epsilon$. Our result builds on a recent line of works showing the existence
of linear sized direct product testers with small soundness by independent
works of Bafna, Lifshitz, and Minzer, and of Dikstein, Dinur, and Lubotzky.
  The main new ingredient in our proof is a technique that embeds a given PCP
construction into a PCP on a prescribed graph, provided that the latter is a
graph underlying a sufficiently good high-dimensional expander. Towards this
end, we use ideas from fault-tolerant distributed computing, and more precisely
from the literature of the almost everywhere agreement problem starting with
the work of Dwork, Peleg, Pippenger, and Upfal (1986). We show that graphs
underlying HDXs admit routing protocols that are tolerant to adversarial edge
corruptions, and in doing so we also improve the state of the art in this line
of work.
  Our PCP construction requires variants of the aforementioned direct product
testers with poly-logarithmic degree. The existence and constructability of
these variants is shown in an appendix by Zhiwei Yun.
