On the NP-Hardness of Rank-Metric Minimum Distance and Syndrome Decoding, at Every Rate

Abstract

In this work, we investigate the hardness of computing minimum distance, as well as the hardness of syndrome decoding, for rank-metric codes. It is known (Gaborit and Zémor, IEEE TIT 2016) that for $\mathbb{F}_{q^m}$-linear rank-metric codes of length $N$, minimum distance when $m=O(N)$ and syndrome decoding when $m=O(N^2)$ are NP-hard under randomized reductions. We obtain a derandomization of these results and a bit more. We show that for any fixed small $\varepsilon>0$, for $\mathbb{F}_{q^m}$-linear rank-metric codes of length $N$ with $m=O(N^{1+\varepsilon})$, the problems of minimum distance, nearest codeword, and syndrome decoding are NP-hard under deterministic reductions. Further, we can prescribe the rate; that is, if we restrict to codes having any fixed rational rate $R\in(0,1)$, then the NP-hardness for all three problems persists. Our main technical contribution is simple, and possibly of independent interest. The reduction of Gaborit and Zémor (IEEE TIT 2016) relies on a distance-preserving embedding of a Hamming-metric code into a rank-metric code, obtained by multiplying its coordinates by a uniformly random vector. We show that we can choose an explicit vector from a slightly larger extension field.