List Decoding Algorithms based on Groebner Bases for General One-Point AG Codes

TL;DR

This paper extends list and unique decoding algorithms based on Gröbner bases from Hermitian and specific AG codes to general one-point algebraic geometry codes, improving efficiency and error correction bounds.

Contribution

It generalizes decoding algorithms to all one-point AG codes under weaker assumptions and enhances their speed and error correction capabilities.

Findings

Extended list decoding to general one-point AG codes.

Improved decoding speed by removing unnecessary steps.

Proved error correction capability equals half the minimum distance.

Abstract

We generalize the list decoding algorithm for Hermitian codes proposed by Lee and O'Sullivan based on Gr\"obner bases to general one-point AG codes, under an assumption weaker than one used by Beelen and Brander. By using the same principle, we also generalize the unique decoding algorithm for one-point AG codes over the Miura-Kamiya $C_{ab}$ curves proposed by Lee, Bras-Amor\'os and O'Sullivan to general one-point AG codes, without any assumption. Finally we extend the latter unique decoding algorithm to list decoding, modify it so that it can be used with the Feng-Rao improved code construction, prove equality between its error correcting capability and half the minimum distance lower bound by Andersen and Geil that has not been done in the original proposal, and remove the unnecessary computational steps so that it can run faster.