Efficient Interpolation-Based Decoding of Reed-Solomon Codes

TL;DR

This paper introduces a novel interpolation-based decoding algorithm for Reed-Solomon codes that leverages FFT and properties of circulant matrices to achieve near-optimal asymptotic complexity for correcting errors.

Contribution

It presents a new decoding algorithm that combines FFT and elimination theory, significantly improving the computational complexity over previous methods.

Findings

Achieves asymptotic complexity of O(t log^2 t) over FFT-friendly fields.

Achieves asymptotic complexity of O(n log^2 n log log n) over arbitrary fields.

Provides the best known asymptotic decoding complexity for 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 $t \leq \lfloor \frac{n-k}{2} \rfloor$ 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.