Unified systems of encoding and decoding for a class of algebraic-geometric codes

We proposed a unified scheme for encoding and decoding by Grobner bases and two-dimensional discrete Fourier transforms (DFTs) for algebraic-geometric codes. In this study, the scheme is improved and reduced for a class of algebraic-geometric codes, generalized norm-trace codes, including Hermitian codes. We often employ the recursive formula from the Grobner bases of locator ideals for some sets of rational points and employ two types of DFTs that satisfy the Fourier inversion formula. We generalize the functioning of generator polynomial for Reed-Solomon codes and develop effective systematic encoding for generalized norm-trace codes. Our encoding has linear-order computational complexity and is highly compatible with the entire decoding that includes location and evaluation of errors.