  In the $(k,h)$-SetCover problem, we are given a collection $\mathcal{S}$ of
sets over a universe $U$, and the goal is to distinguish between the case that
$\mathcal{S}$ contains $k$ sets which cover $U$, from the case that at least
$h$ sets in $\mathcal{S}$ are needed to cover $U$. Lin (ICALP'19) recently
showed a gap creating reduction from the $(k,k+1)$-SetCover problem on universe
of size $O_k(\log |\mathcal{S}|)$ to the
$\left(k,\sqrt[k]{\frac{\log|\mathcal{S}|}{\log\log |\mathcal{S}|}}\cdot
k\right)$-SetCover problem on universe of size $|\mathcal{S}|$. In this paper,
we prove a more scalable version of his result: given any error correcting code
$C$ over alphabet $[q]$, rate $\rho$, and relative distance $\delta$, we use
$C$ to create a reduction from the $(k,k+1)$-SetCover problem on universe $U$
to the $\left(k,\sqrt[2k]{\frac{2}{1-\delta}}\right)$-SetCover problem on
universe of size $\frac{\log|\mathcal{S}|}{\rho}\cdot|U|^{q^k}$.
  Lin established his result by composing the input SetCover instance (that has
no gap) with a special threshold graph constructed from extremal combinatorial
object called universal sets, resulting in a final SetCover instance with gap.
Our reduction follows along the exact same lines, except that we generate the
threshold graphs specified by Lin simply using the basic properties of the
error correcting code $C$.
  We use the same threshold graphs mentioned above to prove inapproximability
results, under W[1]$\neq$FPT and ETH, for the $k$-MaxCover problem introduced
by Chalermsook et al. (SICOMP'20). Our inapproximaiblity results match the
bounds obtained by Karthik et al. (JACM'19), although their proof framework is
very different, and involves generalization of the distributed PCP framework.
Prior to this work, it was not clear how to adopt the proof strategy of Lin to
prove inapproximability results for $k$-MaxCover.
