Determination of error values for decoding hermitian codes with the inverse affine fourier transform
With the knowledge of the syndromes S-a,S-b, 0 less than or equal to a, b less than or equal to 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 q(2)-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 hat; almost the same computation complexity of the conventional discrete Fourier transform, requires the knowledge of syndromes S-a,S-b,S- 0 less than or equal to a, b less than or equal to q - 2, and three more extended syndromes S-q-1,S-q-1,S- S0,(q-1),S-q-1,S-0.
error value hermitian code affine fourier transform algebraic-geometric code