Decoding of differential AG codes

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May 2016, 10(2): 307-319. doi: 10.3934/amc.2016007

Decoding of differential AG codes

Kwankyu Lee ^1,

1.	Department of Mathematics Education, Chosun University, Gwangju 61452, South Korea

Received May 2014 Published April 2016

Abstract
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The interpolation-based decoding that was developed for general evaluation AG codes is shown to be equally applicable to general differential AG codes. A performance analysis of the decoding algorithm, which is parallel to that of its companion algorithm, is reported. In particular, the decoding capacities of evaluation AG codes and differential AG codes are seen to be interrelated symmetrically. As an interesting special case, a decoding algorithm for classical Goppa codes is presented.

Keywords: numerical semigroup., decoding algorithm, Differential AG code, Gröbner basis, interpolation.

Mathematics Subject Classification: Primary: 94B35, 94B27; Secondary: 94B6.

Citation: Kwankyu Lee. Decoding of differential AG codes. Advances in Mathematics of Communications, 2016, 10 (2) : 307-319. doi: 10.3934/amc.2016007

References:

[1]	P. Beelen and T. Høholdt, The decoding of algebraic geometry codes, in Advances in Algebraic Geometry Codes, World Sci. Publ., 2008, 49-98. doi: 10.1142/9789812794017_0002.
[2]	I. M. Duursma, Majority coset decoding, IEEE Trans. Inf. Theory, 39 (1993), 1067-1070. doi: 10.1109/18.256518.
[3]	G. L. Feng and T. T. N. Rao, Decoding algebraic-geometric codes up to the designed minimum distance, IEEE Trans. Inf. Theory, 39 (1993), 37-45. doi: 10.1109/18.179340.
[4]	O. Geil, R. Matsumoto and D. Ruano, List decoding algorithms based on Gröbner bases for general one-point AG codes, in Proc. IEEE Int. Symp. Inf. Theory, 2012, 86-90.
[5]	O. Geil, R. Matsumoto and D. Ruano, Feng-Rao decoding of primary codes, Finite Fields Appl., 23 (2013), 35-52. doi: 10.1016/j.ffa.2013.03.005.
[6]	O. Geil, C. Munuera, D. Ruano and F. Torres, On the order bounds for one-point AG codes, Adv. Math. Commun., 5 (2011), 489-504. doi: 10.3934/amc.2011.5.489.
[7]	V. D. Goppa, Codes on algebraic curves, Sov. Math. Dokl., 24 (1981), 170-172.
[8]	T. Høholdt, J. H. van Lint and R. Pellikaan, Algebraic geometry of codes, in Handbook of Coding Theory, North-Holland, 1998, 871-961.
[9]	K. Lee, Bounds for generalized Hamming weights of general AG codes, Finite Fields Appl., 34 (2015), 265-279. doi: 10.1016/j.ffa.2015.02.006.
[10]	K. Lee, M. Bras-Amorós and M. E. O'Sullivan, Unique decoding of general AG codes, IEEE Trans. Inf. Theory, 60 (2014), 2038-2053. doi: 10.1109/TIT.2014.2306816.
[11]	S. Sakata, H. E. Jensen and T. Høholdt, Generalized Berlekamp-Massey decoding of algebraic-geometric codes up to half the Feng-Rao bound, IEEE Trans. Inf. Theory, 41 (1995), 1762-1768. doi: 10.1109/18.476248.
[12]	H. Stichtenoth, Algebraic Function Fields and Codes, 2nd edition, Springer-Verlag, 2009.

show all references

References:

[1]	P. Beelen and T. Høholdt, The decoding of algebraic geometry codes, in Advances in Algebraic Geometry Codes, World Sci. Publ., 2008, 49-98. doi: 10.1142/9789812794017_0002.
[2]	I. M. Duursma, Majority coset decoding, IEEE Trans. Inf. Theory, 39 (1993), 1067-1070. doi: 10.1109/18.256518.
[3]	G. L. Feng and T. T. N. Rao, Decoding algebraic-geometric codes up to the designed minimum distance, IEEE Trans. Inf. Theory, 39 (1993), 37-45. doi: 10.1109/18.179340.
[4]	O. Geil, R. Matsumoto and D. Ruano, List decoding algorithms based on Gröbner bases for general one-point AG codes, in Proc. IEEE Int. Symp. Inf. Theory, 2012, 86-90.
[5]	O. Geil, R. Matsumoto and D. Ruano, Feng-Rao decoding of primary codes, Finite Fields Appl., 23 (2013), 35-52. doi: 10.1016/j.ffa.2013.03.005.
[6]	O. Geil, C. Munuera, D. Ruano and F. Torres, On the order bounds for one-point AG codes, Adv. Math. Commun., 5 (2011), 489-504. doi: 10.3934/amc.2011.5.489.
[7]	V. D. Goppa, Codes on algebraic curves, Sov. Math. Dokl., 24 (1981), 170-172.
[8]	T. Høholdt, J. H. van Lint and R. Pellikaan, Algebraic geometry of codes, in Handbook of Coding Theory, North-Holland, 1998, 871-961.
[9]	K. Lee, Bounds for generalized Hamming weights of general AG codes, Finite Fields Appl., 34 (2015), 265-279. doi: 10.1016/j.ffa.2015.02.006.
[10]	K. Lee, M. Bras-Amorós and M. E. O'Sullivan, Unique decoding of general AG codes, IEEE Trans. Inf. Theory, 60 (2014), 2038-2053. doi: 10.1109/TIT.2014.2306816.
[11]	S. Sakata, H. E. Jensen and T. Høholdt, Generalized Berlekamp-Massey decoding of algebraic-geometric codes up to half the Feng-Rao bound, IEEE Trans. Inf. Theory, 41 (1995), 1762-1768. doi: 10.1109/18.476248.
[12]	H. Stichtenoth, Algebraic Function Fields and Codes, 2nd edition, Springer-Verlag, 2009.

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