FFT Algorithm for Binary Extension Finite Fields and Its Application to Reed–Solomon Codes

Embargo End Date
2018-08-15

Type
Article

Authors
Lin, Sian Jheng
Al-Naffouri, Tareq Y.
Han, Yunghsiang S.

KAUST Department
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Electrical Engineering Program

Online Publication Date
2016-08-15

Print Publication Date
2016-10

Date
2016-08-15

Abstract
Recently, a new polynomial basis over binary extension fields was proposed, such that the fast Fourier transform (FFT) over such fields can be computed in the complexity of order O(n lg(n)), where n is the number of points evaluated in FFT. In this paper, we reformulate this FFT algorithm, such that it can be easier understood and be extended to develop frequency-domain decoding algorithms for (n = 2(m), k) systematic Reed-Solomon (RS) codes over F-2m, m is an element of Z(+), with n-k a power of two. 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 O(n lg(n-k)+(n-k)lg(2)(n-k)), improving upon the best currently available decoding complexity O(n lg(2)(n) lg lg(n)), and reaching the best known complexity bound that was established by Justesen in 1976. However, Justesen's approach is only for the codes over some specific fields, which can apply Cooley-Tukey FFTs. 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 over F-216.

Citation
Lin S-J, Al-Naffouri TY, Han YS (2016) FFT Algorithm for Binary Extension Finite Fields and Its Application to Reed–Solomon Codes. IEEE Transactions on Information Theory 62: 5343–5358. Available: http://dx.doi.org/10.1109/TIT.2016.2600417.

Acknowledgements
This work was supported in part by the CAS Pioneer Hundred Talents Program and in part by the National Science of Council of Taiwan under Grant NSC 102-2221-E-011-006-MY3 and Grant NSC 101-2221-E -011-069-MY3.

Publisher
Institute of Electrical and Electronics Engineers (IEEE)

Journal
IEEE Transactions on Information Theory

DOI
10.1109/TIT.2016.2600417

arXiv
1503.05761

Additional Links
http://ieeexplore.ieee.org/document/7543456/http://arxiv.org/pdf/1503.05761

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