Decoding up to 4 errors in Hyperbolic-like Abelian Codes by the Sakata Algorithm

TL;DR

This paper enhances the Sakata algorithm for decoding hyperbolic-like abelian codes, enabling correction of up to four errors by optimizing syndrome index selection and improving decoding efficiency.

Contribution

It introduces a refined framework for locator decoding tailored to abelian codes and identifies key syndrome indexes to streamline the decoding process.

Findings

Successful decoding of up to 4 errors in hyperbolic-like abelian codes.

Reduction of syndrome table indexes needed for decoding.

Efficient termination criterion using Groebner basis computation.

Abstract

We deal with two problems related with the use of the Sakata's algorithm in a specific class of bivariate codes. The first one is to improve the general framework of locator decoding in order to apply it on such abelian codes. The second one is to find a set of indexes oF the syndrome table such that no other syndrome contributes to implement the BMSa and, moreover, any of them may be ignored \textit{a priori}. In addition, the implementation on those indexes is sufficient to get the Groebner basis; that is, it is also a termination criterion.