List decoding algorithm based on voting in Gröbner bases for general one-point AG codes

Abstract

We generalize the unique decoding algorithm for one-point AG codes over the Miura-Kamiya C ab curves proposed by Lee, Bras-Amorós 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.(2006) studied the error-correcting capability of the Feng-Rao (Feng and Rao, 1993) or the BMS algorithm (Sakata et al., 1995a,b) with majority voting beyond half the designed distance that are applicable to the dual one-point codes.There was room for improvements in the original result (Lee et al., 2012), namely, (a) they have not clarified the relation between its error-correcting capability and existing minimum distance lower bounds except for the one-point Hermitian codes, (b) they have not analyzed the computational complexity, (c) they assumed that the maximum pole order used for code construction is less than the code length, and (d) they have not shown how to use the method with the Feng-Rao improved code construction (Feng and Rao, 1995). We shall (1) prove that the error-correcting capability of the original proposal is always equal to half of the bound in Andersen and Geil (2008) for the minimum distance of one-point primal codes (Proposition 7), (2) generalize their algorithm to work with any one-point AG codes, (3) modify their algorithm to a list decoding algorithm, (4) remove the assumptions (c) and (d) above, (5) remove unnecessary computational steps from the original proposal, (6) analyze the computational complexity in terms of the number of multiplications and divisions in the finite field. We remark that a generalization of Lee et al. (2012) to arbitrary primal AG code is also reported in Lee et al. (2014) using a similar idea in this paper reported earlier as a conference paper (Geil et al., 2012).The proposed algorithm is implemented on the Singular computer algebra system (Decker et al., 2011), and we verified that...