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

TL;DR

This paper introduces an elementary deterministic reduction from SAT to the Minimum Distance of Code Problem over any finite field, achieving constant factor hardness even for asymptotically good codes.

Contribution

It provides a simple, deterministic, and elementary reduction extending previous randomized and complex reductions, applicable to a broad class of codes.

Findings

Achieves constant factor hardness for asymptotically good codes

Extends reduction to any finite field

Simplifies the reduction process compared to prior complex methods

Abstract

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 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.