  In this paper we present a new gap-creating randomized self-reduction for
parameterized Maximum Likelihood Decoding problem over $\mathbb{F}_p$
($k$-MLD$_p$). The reduction takes a $k$-MLD$_p$ instance with $k\cdot n$
vectors as input, runs in time $f(k)n^{O(1)}$ for some computable function $f$,
outputs a $(3/2-\varepsilon)$-Gap-$k'$-MLD$_p$ instance for any
$\varepsilon>0$, where $k'=O(k^2\log k)$. Using this reduction, we show that
assuming the randomized Exponential Time Hypothesis (ETH), no algorithms can
approximate $k$-MLD$_p$ (and therefore its dual problem $k$-NCP$_p$) within
factor $(3/2-\varepsilon)$ in $f(k)\cdot n^{o(\sqrt{k/\log k})}$ time for any
$\varepsilon>0$.
  We then use reduction by Bhattacharyya, Ghoshal, Karthik and Manurangsi
(ICALP 2018) to amplify the $(3/2-\varepsilon)$-gap to any constant. As a
result, we show that assuming ETH, no algorithms can approximate $k$-NCP$_p$
and $k$-MDP$_p$ within $\gamma$-factor in $f(k)n^{o(k^{\varepsilon_\gamma})}$
time for some constant $\varepsilon_\gamma>0$. Combining with the
gap-preserving reduction by Bennett, Cheraghchi, Guruswami and Ribeiro (STOC
2023), we also obtain similar lower bounds for $k$-MDP$_p$, $k$-CVP$_p$ and
$k$-SVP$_p$.
  These results improve upon the previous $f(k)n^{\Omega(\mathsf{poly} \log
k)}$ lower bounds for these problems under ETH using reductions by
Bhattacharyya et al. (J.ACM 2021) and Bennett et al. (STOC 2023).
