Improved decoding of affine-variety codes

TL;DR

This paper introduces new decoding techniques for affine-variety codes using multidimensional error locator polynomials, proving their existence for all correctable affine-variety codes and extending the theory of stratified ideals.

Contribution

It generalizes error locator polynomials to multiple variables and points with multiplicities, providing new decoding methods for affine-variety codes.

Findings

Existence of multidimensional error locator polynomials for all correctable affine-variety codes

Development of decoding techniques based on geometric approaches

Extension of stratified ideals to multiple variables and multiplicities

Abstract

General error locator polynomials are polynomials able to decode any correctable syndrome for a given linear code. Such polynomials are known to exist for all cyclic codes and for a large class of linear codes. We provide some decoding techniques for affine-variety codes using some multidimensional extensions of general error locator polynomials. We prove the existence of such polynomials for any correctable affine-variety code and hence for any linear code. We propose two main different approaches, that depend on the underlying geometry. We compute some interesting cases, including Hermitian codes. To prove our coding theory results, we develop a theory for special classes of zero-dimensional ideals, that can be considered generalizations of stratified ideals. Our improvement with respect to stratified ideals is twofold: we generalize from one variable to many variables and we introduce points with multiplicities.