AI 中文总结
针对远程点问题，本文在有理数域上给出达到最优远程度n-k的确定性多项式时间算法，并在有限域上改进了已知结果，达到Omega(n/max{k,log n} log n)的远程度。
AI 中文摘要
远程点问题（RPP）是一个算法问题，给定一个维度为 $k$ 的线性子空间 $L \subseteq \mathbb{F}^n$，要求确定性地找到一个向量 $v \in \mathbb{F}^n$，使其与 $L$ 的汉明距离尽可能远。该问题由 Alon、Panigrahy 和 Yekhanin [APY09] 提出，部分动机源于用于证明电路下界的矩阵刚性方法。若一个算法能找到与 $L$ 的汉明距离至少为 $d$ 的向量 $v$，则称该算法达到远程度 $d$。我们观察到，在有理数域上，该问题存在一个确定性多项式时间算法，能达到最优远程度 $n-k$。在有限域上，我们（适度）改进了 Alon、Panigrahy 和 Yekhanin [APY09] 的结果，并给出一个算法，能达到远程度 $\Omega\left(\frac{n}{\max\{k, \log n\}} \log n\right)$。
英文摘要
The Remote Point Problem (RPP) is an algorithmic problem that asks, given a linear subspace $L \subseteq \mathbb{F}^n$ of dimension $k$, to deterministically find a vector $v \in \mathbb{F}^n$ far in Hamming distance from $L$. This problem was introduced by Alon, Panigrahy and Yekhanin [APY09], motivated in part by the matrix rigidity approach for proving circuit lower bounds. An algorithm is said to achieve remoteness $d$ if it finds a vector $v$ whose Hamming distance from $L$ is at least $d$. We observe that over the rational numbers, the problem admits a deterministic polynomial-time algorithm that achieves optimal remoteness $n-k$. Over finite fields, we obtain a (modest) improvement of a result of Alon, Panigrahy and Yekhanin [APY09], and give an algorithm that achieves remoteness $Ω\left(\frac{n}{\max\{k, \log n\}} \log n\right)$.
