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

作者：Austrin Per^*; Khot Subhash

来源：IEEE Transactions on Information Theory, 2014, 60(10): 6636-6645.

DOI：10.1109/TIT.2014.2340869

• Linear code
• computational complexity

We present a simple deterministic gap-preserving reduction from SAT to the minimum distance of code problem over F-2. We also show how to extend the reduction to work over any fixed 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 nontrivial coding theoretic constructions, whereas our reduction is elementary. As an additional feature, our reduction gives hardness within a constant factor even for asymptotically good codes, i.e., having constant positive rate and relative distance. Previously, it was not known how to achieve a deterministic reduction for such codes.

• 出版日期2014-10