The inapproximability of lattice and coding problems with preprocessing

We prove that the closest vector problem with preprocessing (CVPP) is NP-hard to approximate within any factor less than /spl radic/5/3. More specifically, we show that there exists a reduction from an NP-hard problem to the approximate closest vector problem such that the lattice depends only on the size of the original problem, and the specific instance is encoded solely, in the target vector. It follows that there are lattices for which the closest vector problem cannot be approximated within factors /spl gamma/ < /spl radic/5/3 in polynomial time, no matter how the lattice is represented, unless NP is equal to P (or NP is contained in P/poly, in case of nonuniform sequences of lattices). The result easily extends to any lp norm, for p /spl ges/ 1, showing that CVPP in the lp norm is hard to approximate within any factor /spl gamma/ < /sup p//spl radic/5/3. As an intermediate step, we establish analogous results for the nearest codeword problem with preprocessing (NCPP), proving that for any finite field GF(q), NCPP over GF(q) is NP-hard to approximate within any factor less than 5/3.