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

TL;DR

This paper introduces a lemma in algebraic coding theory that facilitates fast, unified encoding and decoding of a specific class of affine variety codes, improving efficiency in error correction processes.

Contribution

The paper presents a new lemma linking vector spaces in algebraic coding, enabling a fast, systematic approach to encoding and decoding affine variety codes.

Findings

Established a key algebraic lemma used in coding theory

Developed a fast, unified encoding and decoding system for affine variety codes

Demonstrated improved efficiency in error and erasure correction

Abstract

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 Grobner basis, are canonically isomorphic, and moreover, the isomorphism is given by the extension through linear feedback shift registers from Grobner 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.