A Simple Deterministic Reduction for the Gap Minimum Distance of Code Problem

We present a simple deterministic gap-preserving reduction from SAT to the Minimum Distance of Code Problem over ${\text{F}}_2$. 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, which was recently derandomized by Cheng and Wan. 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.