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The fast decoding of Reed-Solomon codes using Fermat transforms (Corresp.)

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It is shown that \sqrt [8]{2} is an element of order 2^{n+4} in GF(F_{n}) , where F_{n}=2^{2^{n}}+1 is a Fermat prime for n=3,4 . Hence it can be used to define a fast Fourier transform (FFT) of as many as 2^{n+4} symbols in GF(F_{n}) . Since \sqrt [8]{2} is a root of unity of order 2^{n+4} in GF(F_{n}) , this transform requires fewer muitiplications than the conventional FFT algorithm. Moreover, as Justesen points out [1], such an FFT can be used to decode certain Reed-Solomon codes. An example of such a transform decoder for the case n=2 , where \sqrt {2} is in GF(F_{2})=GF(17) , is given.

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Information Theory, IEEE Transactions on  (Volume:24 ,  Issue: 4 )