We show that, unless NP\subseteqDTIME(2^{poly\log (n)}), the closest vector problem with pre-processing, for \ell \rho norm for any p \ge 1, is hard to approximate within a factor of (\log n)^{1/p - \ell } for any \varepsilon > 0. This improves the previous best factor of 3^{1/p} - \varepsilon due to Regev [19]. 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 - \varepsilon } for any \varepsilon > 0.