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

摘要

In this paper, a lemma in algebraic coding theory is established, which isfrequently appeared in the encoding and decoding for algebraic codes such asReed-Solomon codes and algebraic geometry codes. This lemma states that twovector spaces, one corresponds to information symbols and the other is indexedby the support of Grobner basis, are canonically isomorphic, and moreover, theisomorphism is given by the extension through linear feedback shift registersfrom Grobner basis and discrete Fourier transforms. Next, the lemma is appliedto fast unified system of encoding and decoding erasures and errors in acertain class of affine variety codes.