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We generalize the unique decoding algorithm for one-point AG codes over theMiura-Kamiya Cab curves proposed by Lee, Bras-Amor\'os and O'Sullivan (2012) togeneral one-point AG codes, without any assumption. We also extend their uniquedecoding algorithm to list decoding, modify it so that it can be used with theFeng-Rao improved code construction, prove equality between its errorcorrecting capability and half the minimum distance lower bound by Andersen andGeil (2008) that has not been done in the original proposal except forone-point Hermitian codes, remove the unnecessary computational steps so thatit can run faster, and analyze its computational complexity in terms ofmultiplications and divisions in the finite field. As a unique decodingalgorithm, the proposed one is empirically and theoretically as fast as the BMSalgorithm for one-point Hermitian codes. As a list decoding algorithm,extensive experiments suggest that it can be much faster for many moderatesize/usual inputs than the algorithm by Beelen and Brander (2010). It should benoted that as a list decoding algorithm the proposed method seems to haveexponential worst-case computational complexity while the previous proposals(Beelen and Brander, 2010; Guruswami and Sudan, 1999) have polynomial ones, andthat the proposed method is expected to be slower than the previous proposalsfor very large/special inputs.

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