  We solve the derandomized direct product testing question in the low
acceptance regime, by constructing new high dimensional expanders that have no
small connected covers. We show that our complexes have swap cocycle expansion,
which allows us to deduce the agreement theorem by relying on previous work.
  Derandomized direct product testing, also known as agreement testing, is the
following problem. Let X be a family of k-element subsets of [n] and let
$\{f_s:s\to\Sigma\}_{s\in X}$ be an ensemble of local functions, each defined
over a subset $s\subset [n]$. Suppose that we run the following so-called
agreement test: choose a random pair of sets $s_1,s_2\in X$ that intersect on
$\sqrt k$ elements, and accept if $f_{s_1},f_{s_2}$ agree on the elements in
$s_1\cap s_2$. We denote the success probability of this test by
$Agr(\{f_s\})$. Given that $Agr(\{f_s\})=\epsilon>0$, is there a global
function $G:[n]\to\Sigma$ such that $f_s = G|_s$ for a non-negligible fraction
of $s\in X$ ?
  We construct a family X of k-subsets of $[n]$ such that $|X| = O(n)$ and such
that it satisfies the low acceptance agreement theorem. Namely,
  $Agr (\{f_s\}) > \epsilon \; \; \longrightarrow$ there is a function
$G:[n]\to\Sigma$ such that $\Pr_s[f_s\overset{0.99}{\approx} G|_s]\geq
poly(\epsilon)$.
  A key idea is to replace the well-studied LSV complexes by symplectic high
dimensional expanders (HDXs). The family X is just the k-faces of the new
symplectic HDXs. The later serve our needs better since their fundamental group
satisfies the congruence subgroup property, which implies that they lack small
covers. We also give a polynomial-time algorithm to construct this family of
symplectic HDXs.
