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.

翻译：我们证明，在任意固定有限域上的线性码中，参数化输入距离界的最小距离问题（MDP）在W[1]-意义下难以在任意常数因子内近似。我们同样证明了整数格中参数化最短向量问题（SVP）的类似结论。具体来说，我们证明：对任意固定$p>1$，ℓₚ范数下的SVP在W[1]-意义下难以在任意常数因子内近似；当$p=1$时，SVP在W[1]-意义下难以在逼近$2$的因子内近似（每种情形均在随机归约下证明困难性）。这些结果回答了Bhattacharyya、Bonnet、Egri、Ghoshal、Karthik C. S.、Lin、Manurangsi和Marx（Journal of the ACM, 2021）关于参数化MDP与SVP复杂性遗留的主要公开问题。对于MDP，他们仅在二元线性码上证明了类似困难性，而一般域的情况未予解决；对于$p>1$的ℓₚ范数SVP，他们证明了在某个依赖$p$的常数因子内难近似，但未证明任意常数因子下的困难性；对于ℓ₁范数下精确SVP的W[1]-困难性，他们同样未予解决。