February  2017, 11(1): 115-122. doi: 10.3934/amc.2017006

Generalized Hamming weights of codes over the $\mathcal{GH}$ curve

Universidade Federal de Uberlândia, Campus Santa Mônica, CEP 38.408-100, Av. Joao Naves de Avila, 2121, Uberlandia-MG, Brazil

Received  June 2015 Revised  February 2016 Published  February 2017

Fund Project: The author would like to thank FAPEMIG by support.

In this work, we studied the generalized Hamming weights of algebraic geometric Goppa codes on the $\mathcal{GH}$ curve. Especially, exact results on the second generalized Hamming weight are given for almost all cases. Furthermore, we apply the results obtained to show an example where the weight hierarchy characterizes the performance of the $\mathcal{GH}$ codes on a noiseless communication channel.

Citation: Alonso sepúlveda Castellanos. Generalized Hamming weights of codes over the $\mathcal{GH}$ curve. Advances in Mathematics of Communications, 2017, 11 (1) : 115-122. doi: 10.3934/amc.2017006
References:
[1]

S. V. Bulygin, Generalized Hermitian codes over GF (2r), IEEE Trans. Inform. Theory, 52 (2006), 4664-4669.  doi: 10.1109/TIT.2006.881831.  Google Scholar

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A. Garcia and H. Stichtenoth, A class of polynomials over finite fields, Finite Fields Appl., 5 (1999), 424-435.  doi: 10.1006/ffta.1999.0261.  Google Scholar

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O. GeilC. MunueraD. Ruano and F. Torres, On the order bounds for one-point AG codes, Adv. Math. Commun., 3 (2011), 489-504.  doi: 10.3934/amc.2011.5.489.  Google Scholar

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V. D. Goppa, Codes associated with divisors, Problems Inform. Transm., 13 (1977), 22-26.   Google Scholar

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P. Heijnen and R. Pellikaan, Generalized Hamming weights of q-ary Reed-Muller codes, IEEE Trans. Inform. Theory, 44 (1998), 181-197.  doi: 10.1109/18.651015.  Google Scholar

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T. Høholdt, J. H. van Lint and R. Pellikaan, Algebraic-geometry codes, in Handbook of Coding Theory, Elsevier, Amsterdam, 1998.  Google Scholar

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P. V. KumarH. Stichtenoth and K. Yang, On the weight hierarchy of geometric Goppa codes, IEEE Trans. Inform. Theory, 40 (1994), 913-920.  doi: 10.1109/18.335903.  Google Scholar

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H. Lange and G. Martens, On the gonality sequence of an algebraic curve, Manuscripta Math., 137 (2012), 457-473.  doi: 10.1007/s00229-011-0475-4.  Google Scholar

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C. Munuera, On the generalized Hamming weights of geometric Goppa codes, IEEE Trans. Inform. Theory, 40 (1994), 2092-2099.  doi: 10.1109/18.340488.  Google Scholar

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C. MunueraA. Sep′ulveda and F. Torres, Castle curves and codes, Adv. Math. Commun., 4 (2009), 399-408.  doi: 10.3934/amc.2009.3.399.  Google Scholar

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C. MunueraA. Sep′ulveda and F. Torres, Generalized Hermitian codes, Des. Codes Cryptogr., 69 (2013), 123-130.  doi: 10.1007/s10623-012-9627-0.  Google Scholar

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C. Munuera and F. Torres, Bounding the trellis state complexity of algebraic geometric codes, Appl. Algebra Engrg. Comm. Comput., 15 (2004), 81-100.  doi: 10.1007/s00200-004-0150-z.  Google Scholar

[13]

W. Olaya León, Pesos de Hamming de Códigos Castillo, Ph. D thesis, University of Valladolid, Spain, 2014. Google Scholar

[14]

L. H. Ozarow and A. D. Wyner, Wire-Tap-channel Ⅱ, AT & T Bell Labs. Tech. J., 63 (1984), 2135-2157.  doi: 10.1002/j.1538-7305.1975.tb02040.x.  Google Scholar

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H. Stichtenoth, Algebraic Function Fields and Codes, Springer-Verlag, New York, 1993.  Google Scholar

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V. K. Wei, Generalized Hamming weights for linear codes, IEEE Trans. Inform. Theory, 37 (1991), 1412-1418.  doi: 10.1109/18.133259.  Google Scholar

show all references

References:
[1]

S. V. Bulygin, Generalized Hermitian codes over GF (2r), IEEE Trans. Inform. Theory, 52 (2006), 4664-4669.  doi: 10.1109/TIT.2006.881831.  Google Scholar

[2]

A. Garcia and H. Stichtenoth, A class of polynomials over finite fields, Finite Fields Appl., 5 (1999), 424-435.  doi: 10.1006/ffta.1999.0261.  Google Scholar

[3]

O. GeilC. MunueraD. Ruano and F. Torres, On the order bounds for one-point AG codes, Adv. Math. Commun., 3 (2011), 489-504.  doi: 10.3934/amc.2011.5.489.  Google Scholar

[4]

V. D. Goppa, Codes associated with divisors, Problems Inform. Transm., 13 (1977), 22-26.   Google Scholar

[5]

P. Heijnen and R. Pellikaan, Generalized Hamming weights of q-ary Reed-Muller codes, IEEE Trans. Inform. Theory, 44 (1998), 181-197.  doi: 10.1109/18.651015.  Google Scholar

[6]

T. Høholdt, J. H. van Lint and R. Pellikaan, Algebraic-geometry codes, in Handbook of Coding Theory, Elsevier, Amsterdam, 1998.  Google Scholar

[7]

P. V. KumarH. Stichtenoth and K. Yang, On the weight hierarchy of geometric Goppa codes, IEEE Trans. Inform. Theory, 40 (1994), 913-920.  doi: 10.1109/18.335903.  Google Scholar

[8]

H. Lange and G. Martens, On the gonality sequence of an algebraic curve, Manuscripta Math., 137 (2012), 457-473.  doi: 10.1007/s00229-011-0475-4.  Google Scholar

[9]

C. Munuera, On the generalized Hamming weights of geometric Goppa codes, IEEE Trans. Inform. Theory, 40 (1994), 2092-2099.  doi: 10.1109/18.340488.  Google Scholar

[10]

C. MunueraA. Sep′ulveda and F. Torres, Castle curves and codes, Adv. Math. Commun., 4 (2009), 399-408.  doi: 10.3934/amc.2009.3.399.  Google Scholar

[11]

C. MunueraA. Sep′ulveda and F. Torres, Generalized Hermitian codes, Des. Codes Cryptogr., 69 (2013), 123-130.  doi: 10.1007/s10623-012-9627-0.  Google Scholar

[12]

C. Munuera and F. Torres, Bounding the trellis state complexity of algebraic geometric codes, Appl. Algebra Engrg. Comm. Comput., 15 (2004), 81-100.  doi: 10.1007/s00200-004-0150-z.  Google Scholar

[13]

W. Olaya León, Pesos de Hamming de Códigos Castillo, Ph. D thesis, University of Valladolid, Spain, 2014. Google Scholar

[14]

L. H. Ozarow and A. D. Wyner, Wire-Tap-channel Ⅱ, AT & T Bell Labs. Tech. J., 63 (1984), 2135-2157.  doi: 10.1002/j.1538-7305.1975.tb02040.x.  Google Scholar

[15]

H. Stichtenoth, Algebraic Function Fields and Codes, Springer-Verlag, New York, 1993.  Google Scholar

[16]

V. K. Wei, Generalized Hamming weights for linear codes, IEEE Trans. Inform. Theory, 37 (1991), 1412-1418.  doi: 10.1109/18.133259.  Google Scholar

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