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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 thelength and $k$ is the dimension. In particular, we employ the fast Fouriertransform (FFT) together with properties of a circulant matrix associated withthe error interpolation polynomial and some known results from eliminationtheory in the decoding process. The asymptotic computational complexity of theproposed 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 finitefields, respectively, achieving the best currently known asymptotic decodingcomplexity, proposed for the same set of parameters.

Efficient Interpolation-Based Decoding of Reed-Solomon Codes