In this paper, we propose a probabilistic algorithm suitable for any linear code $C$ to determine whether a given vector $\mathbf{x}$ belongs to $ C$. The algorithm achieves $O(n\log n)$ time complexity, $ O(n^2)$ space complexity and with an error probability less than $1/\mathrm{poly}(n)$ in the asymptotic sense.

翻译：本文提出了一种适用于任意线性码$C$的概率算法，用于判定给定向量$\mathbf{x}$是否属于$C$。该算法的时间复杂度为$O(n\log n)$，空间复杂度为$O(n^2)$，且渐近意义上的错误概率小于$1/\mathrm{poly}(n)$。