Unified system of encoding and decoding erasures and errors for algebraic geometry codes

In this paper, a fundamental lemma in algebraic coding theory is established, which is frequently appeared in the encoding and decoding for algebraic codes such as Reed-Solomon and algebraic geometry codes. This lemma states that two vector spaces, one corresponds to information symbols and the other is indexed by the support of Gröbner basis, are canonically isomorphic, and moreover, the isomorphism is given in terms of extension by linear feedback shift registers from Gröbner basis and discrete Fourier transforms. Next, we apply the lemma to unified system of encoding and decoding erasure-errors in algebraic geometry codes. Finally, we comment on an improved bound for the generic erasure-error correcting capabilities.