Fast Erasure-and-Error Decoding and Systematic Encoding of a Class of Affine Variety Codes

In this paper, a lemma in algebraic coding theory is established, which is frequently appeared in the encoding and decoding for algebraic codes such as Reed–Solomon codes 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 by the extension through linear feedback shift registers from Gröbner basis and discrete Fourier transforms. Next, the lemma is applied to fast unified system of encoding and decoding erasures and errors in a certain class of affine variety codes.