Efficient Frequency-Domain Decoding Algorithms for Reed-Solomon Codes.
Sian-Jheng Lin, Tareq Y. Al-Naffouri, Yunghsiang Sam Han · arXiv (Cornell University) · 2015
This work develops frequency-domain decoding algorithms for $(n=2^m,k)$ systematic Reed-Solomon (RS) codes over fields $\mathbb{F}_{2^m},m\in \mathbb{Z}^+$, where $n-k$ is a power of two. The proposed algorithms are based on a new polynomial basis with a fast Fourier transform with computational complexity of order $\mathcal{O}(n\lg(n))$. First, the basis of syndrome polynomials is reformulated in the decoding procedure so that the new transforms can be applied to the decoding procedure. A fast extended Euclidean algorithm is developed to determine the error locator polynomial. The computational complexity of the proposed decoding algorithm is $\mathcal{O}(n\lg(n-k)+(n-k)\lg^2(n-k))$, improving upon the best currently available decoding complexity of $\mathcal{O}(n\lg^2(n)\lg\lg(n))$ and reaching the best known complexity bound that was established by Justesen in 1976, whose approach is for RS codes that operate only on some specified finite fields. As revealed by the computer simulations, the proposed decoding algorithm is $50$ times faster than the conventional one for the $(2^{16},2^{15})$ RS code.