List decoding algorithm based on voting in Gröbner bases for general one-point AG codes. (March 2017)

• Record Type: Journal Article
• Authors: Matsumoto, Ryutaroh; Ruano, Diego; Geil, Olav
• Is Part Of: Journal of symbolic computation. Volume 79(2017)Part 2
• Page Start: 384
• Page End: 410
• Publication Date: 2017-03
• Subjects: Algebraic geometry code -- Gröbner basis -- List decoding; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2016.02.015
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: We generalize the unique decoding algorithm for one-point AG codes over the Miura–Kamiya C a b curves proposed byLee 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 byAndersen 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 byBeelen 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.