We propose a new interpolation-based error decoding algorithm for $(n,k)$ Reed-Solomon (RS) codes over a finite field of size $q$, where $n=q-1$ is the length and $k$ is the dimension. In particular, we employ the fast Fourier transform (FFT) together with properties of a circulant matrix associated with the error interpolation polynomial and some known results from elimination theory in the decoding process. The asymptotic computational complexity of the proposed algorithm for correcting any $t \leq \lfloor \frac{n-k}{2} \rfloor$ errors in an $(n,k)$ RS code is of order $\mathcal{O}(t\log^2 t)$ and $\mathcal{O}(n\log^2 n \log\log n)$ over FFT-friendly and arbitrary finite fields, respectively, achieving the best currently known asymptotic decoding complexity, proposed for the same set of parameters.

翻译：我们提出了一种新的基于插值的纠错译码算法，适用于大小为$q$的有限域上的$(n,k)$里德-所罗门（RS）码，其中$n=q-1$为码长，$k$为维数。具体而言，我们在译码过程中利用快速傅里叶变换（FFT）及与纠错插值多项式相关的循环矩阵性质，并结合消去理论中的已知结果。所提算法用于纠正$(n,k)$ RS码中任意$t \leq \lfloor \frac{n-k}{2} \rfloor$个错误时，在适合FFT的有限域上渐近计算复杂度为$\mathcal{O}(t\log^2 t)$，在任意有限域上为$\mathcal{O}(n\log^2 n \log\log n)$，针对相同参数集实现了当前已知的最优渐近译码复杂度。