Efficient Interpolation-Based Decoding of Reed-Solomon Codes

Abstract

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.

Presented at the 2023 IEEE International Symposium on Information Theory (ISIT)