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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

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.  Google Scholar

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I. M. Duursma, Majority coset decoding, IEEE Trans. Inf. Theory, 39 (1993), 1067-1070. doi: 10.1109/18.256518.  Google Scholar

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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.  Google Scholar

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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. Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[7]

V. D. Goppa, Codes on algebraic curves, Sov. Math. Dokl., 24 (1981), 170-172.  Google Scholar

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T. Høholdt, J. H. van Lint and R. Pellikaan, Algebraic geometry of codes, in Handbook of Coding Theory, North-Holland, 1998, 871-961.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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H. Stichtenoth, Algebraic Function Fields and Codes, 2nd edition, Springer-Verlag, 2009.  Google Scholar

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.  Google Scholar

[2]

I. M. Duursma, Majority coset decoding, IEEE Trans. Inf. Theory, 39 (1993), 1067-1070. doi: 10.1109/18.256518.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

V. D. Goppa, Codes on algebraic curves, Sov. Math. Dokl., 24 (1981), 170-172.  Google Scholar

[8]

T. Høholdt, J. H. van Lint and R. Pellikaan, Algebraic geometry of codes, in Handbook of Coding Theory, North-Holland, 1998, 871-961.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

H. Stichtenoth, Algebraic Function Fields and Codes, 2nd edition, Springer-Verlag, 2009.  Google Scholar

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