Systolic array implementations of modified Gaussian eliminations for the decoding of an (n, n-2t) RS code, including the Hong-Vetterli algorithm and the FIA proposed by Feng and Tzeng, are designed in this paper. These modified Gaussian eliminations are more easily understanding than the classical Berlekamp-Massey algorithm and, in addition, are efficient to decode RS codes for small e or e <

With the knowledge of the syndromes Sa,b, 0a,b q-2, the exact error values cannot be determined by using the conventional (q-1)2-point discrete Fourier transform in the decoding of a plane algebraic-geometric code over GF(q). In this letter, the inverse q-point 1-dimensional and q2-point 2-dimensional affine Fourier transform over GF(q) are presented to be used to retrieve the actual error values, but it requires much computation efforts. For saving computation complexity, a modification of the affine Fourier transform is derived by using the property of the rational points of the plane Hermitian curve. The modified transform, which has almost the same computation complexity of the conventional discrete Fourier transform, requires the knowledge of syndromes Sa,b, 0 a,b q-2, and three more extended syndromes Sq-1,q-1, S0,q-1, Sq-1,0.