Hardness of approximating the closest vector problem with pre-processing

We show that, unless NP⊆DTIME(2^{poly log(n)}) the closest vector problem with pre-processing, for ℓ_p norm for any p ≥ 1, is hard to approximate within a factor of (log n)¹p - ε/´ /P for any ε > 0. This improves the previous best factor of 3¹p/ - ε due to Regev (2004). Our results also imply that under the same complexity assumption, the nearest codeword problem with pre-processing is hard to approximate within a factor of (log n)^{1 - ε}´ for any ε > 0.