Probabilistic verification algorithm for linear codes

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 $1\mathrm{/}\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}(n)$ in the asymptotic sense.