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arXiv · 2307.00891

Efficient Interpolation-Based Decoding of Reed-Solomon Codes

Wrya K. Kadir·Hsuan-Yin Lin·Eirik Rosnes
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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 $t \leq \lfloor \frac{n-k}{2} \rfloor$ 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.

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Wrya K. Kadir, Hsuan-Yin Lin, Eirik Rosnes. 2023-07-03. Efficient Interpolation-Based Decoding of Reed-Solomon Codes. https://arxiv.org/abs/2307.00891

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