Efficient Interpolation-Based Decoding of Reed-Solomon Codes

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 $t\le \lfloor \frac{n-k}{2}\rfloor$ 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.