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).

翻译：本文提出了一种新的间隙创造型随机自归约，针对$\mathbb{F}_p$上的参数化最大似然解码问题（$k$-MLD$_p$）。该归约以包含$k\cdot n$个向量的$k$-MLD$_p$实例为输入，在$f(k)n^{O(1)}$时间内运行（$f$为可计算函数），对任意$\varepsilon>0$输出一个$(3/2-\varepsilon)$-Gap-$k'$-MLD$_p$实例，其中$k'=O(k^2\log k)$。利用此归约，我们证明：在随机指数时间假设（ETH）下，对于任意$\varepsilon>0$，不存在算法能在$f(k)\cdot n^{o(\sqrt{k/\log k})}$时间内以$(3/2-\varepsilon)$因子近似$k$-MLD$_p$（及其对偶问题$k$-NCP$_p$）。随后，我们采用Bhattacharyya、Ghoshal、Karthik和Manurangsi（ICALP 2018）的归约将$(3/2-\varepsilon)$间隙放大至任意常数。由此证明：在ETH假设下，对某些常数$\varepsilon_\gamma>0$，不存在算法能在$f(k)n^{o(k^{\varepsilon_\gamma})}$时间内以$\gamma$因子近似$k$-NCP$_p$和$k$-MDP$_p$。结合Bennett、Cheraghchi、Guruswami和Ribeiro（STOC 2023）的保间隙归约，我们进一步得到$k$-MDP$_p$、$k$-CVP$_p$和$k$-SVP$_p$的类似下界。这些结果改进了先前利用Bhattacharyya等人（J.ACM 2021）和Bennett等人（STOC 2023）归约所获得的ETH下$f(k)n^{\Omega(\mathsf{poly} \log k)}$下界。