List Decoding Algorithm based on Voting in Groebner Bases for General One-Point AG Codes

TL;DR

This paper introduces a generalized list decoding algorithm for one-point algebraic geometry codes using voting in Groebner bases, improving speed and extending applicability beyond Hermitian codes.

Contribution

It extends the unique decoding algorithm to general one-point AG codes, incorporates list decoding, and analyzes its complexity and performance.

Findings

Empirically as fast as BMS algorithm for Hermitian codes

Can be much faster than Beelen and Brander's algorithm for moderate inputs

Has exponential worst-case complexity, slower for very large inputs

Abstract

We generalize the unique decoding algorithm for one-point AG codes over the Miura-Kamiya Cab curves proposed by Lee, Bras-Amor\'os and O'Sullivan (2012) to general one-point AG codes, without any assumption. We also extend their 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 (2008) that has not been done in the original proposal except for one-point Hermitian codes, remove the unnecessary computational steps so that it can run faster, and analyze its computational complexity in terms of multiplications and divisions in the finite field. As a unique decoding algorithm, the proposed one is empirically and theoretically as fast as the BMS algorithm for one-point Hermitian codes. As a list decoding algorithm, extensive experiments suggest that it can be much faster for many moderate size/usual inputs than the algorithm by Beelen and Brander (2010). It should be noted that as a list decoding algorithm the proposed method seems to have exponential worst-case computational complexity while the previous proposals (Beelen and Brander, 2010; Guruswami and Sudan, 1999) have polynomial ones, and that the proposed method is expected to be slower than the previous proposals for very large/special inputs.