Given a linear subspace of $n \times n$ matrices over $\mathbb F_{2^r}$ that is promised to contain a matrix of rank $1$, we prove that it is hard to find a matrix of rank $n^{o(1/\log \log n)}$, assuming NP doesn't have sub-exponential algorithms. In addition to being a basic problem, the hardness of this problem, even for the exact version, drove recent PCP-free inapproximability results for minimum distance and shortest vector problems concerning codes and lattices. The proof combines the concept of superposition soundness introduced by Khot and Saket with moment matrices. To produce a rank-gap of $1$ vs. $k$, the reduction runs in time $n^{O(\log k)}$. We also give another moment-matrix-based construction which runs in time $n^{O(k)}$ but works for any finite field $\mathbb F_q$.

翻译：给定一个在 $\mathbb F_{2^r}$ 上的 $n \times n$ 矩阵所构成的线性子空间，且该子空间被承诺包含一个秩为 $1$ 的矩阵，我们证明，在假设NP不存在次指数算法的情况下，难以找到一个秩为 $n^{o(1/\log \log n)}$ 的矩阵。除了本身是一个基本问题外，该问题的困难性——即使是在精确版本下——推动了近期关于编码和格的最小距离与最短向量问题的无PCP不可逼近性结果。本证明结合了Khot和Saket引入的叠加可靠性概念与矩矩阵。为了产生 $1$ 与 $k$ 之间的秩差距，归约的运行时间为 $n^{O(\log k)}$。我们同时给出另一种基于矩矩阵的构造，其运行时间为 $n^{O(k)}$，但适用于任意有限域 $\mathbb F_q$。