We present a simple deterministic gap-preserving reduction from SAT to the Minimum Distance of Code Problem over GF(2). We also show how to extend the reduction to work over any finite field. Previously a randomized reduction was known due to Dumer, Micciancio, and Sudan [1999], which was recently derandomized by Cheng and Wan [2009]. These reductions rely on highly non-trivial coding theoretic constructions whereas our reduction is elementary.

As an additional feature, our reduction gives a constant factor hardness even for asymptotically good codes, i.e., having constant rate and relative distance. Previously it was not known how to achieve deterministic reductions for such codes.