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

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• 作者：Ryutaroh Matsumoto^a ; ^{ryutaroh@rmatsumoto.org" class="auth_mail" title="E-mail the corresponding author} ; Diego Ruano^b ; ^{diego@math.aau.dk" class="auth_mail" title="E-mail the corresponding author} ; Olav Geil^b ; ^{olav@math.aau.dk" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Algebraic geometry code ; Grö ; bner basis ; List decoding
• 刊名：Journal of Symbolic Computation
• 出版年：2017
• 出版时间：March-April 2017
• 年：2017
• 卷：79
• 期：part_P2
• 页码：384-410
• 全文大小：646 K

文摘

We generalize the unique decoding algorithm for one-point AG codes over the Miura–Kamiya 16000316&_mathId=si1.gif&_user=111111111&_pii=S0747717116000316&_rdoc=1&_issn=07477171&md5=90c146eedb3ae30b9652ba3b16729726" title="Click to view the MathML source">C_ab$C_{ab}$ curves proposed by Lee et al. (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 ( and ) have polynomial ones, and that the proposed method is expected to be slower than the previous proposals for very large/special inputs.