Fast transform for decoding both errors and erasures of Reed-Solomon codes over GF(2m) for 8≤m≤10

In this letter, it is shown that a fast, prime-factor discrete Fourier transform (DFT) algorithm can be modified to compute Fourier-like transforms of long sequences of 2^m-1 points over GF(2^m), where 8≤m≤10. Using these transforms, together with the Berlekamp-Massey algorithm, the complexity of the transform-domain decoder for correcting both errors and erasures of the Reed-Solomon codes of block length 2^m-1 over GF(2^m) for 8≤m≤10 is reduced substantially from the previous time-domain decoder. A computer simulation verifies these new results.