  We prove that the Minimum Distance Problem (MDP) on linear codes over any
fixed finite field and parameterized by the input distance bound is W[1]-hard
to approximate within any constant factor. We also prove analogous results for
the parameterized Shortest Vector Problem (SVP) on integer lattices.
Specifically, we prove that SVP in the $\ell_p$ norm is W[1]-hard to
approximate within any constant factor for any fixed $p >1$ and W[1]-hard to
approximate within a factor approaching $2$ for $p=1$. (We show hardness under
randomized reductions in each case.)
  These results answer the main questions left open (and explicitly posed) by
Bhattacharyya, Bonnet, Egri, Ghoshal, Karthik C. S., Lin, Manurangsi, and Marx
(Journal of the ACM, 2021) on the complexity of parameterized MDP and SVP. For
MDP, they established similar hardness for binary linear codes and left the
case of general fields open. For SVP in $\ell_p$ norms with $p > 1$, they
showed inapproximability within some constant factor (depending on $p$) and
left open showing such hardness for arbitrary constant factors. They also left
open showing W[1]-hardness even of exact SVP in the $\ell_1$ norm.
