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.

[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.

[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.

[4]

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

[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.

[6]

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

[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.

[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.

[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.

[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.

[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.

[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.

[13]

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

[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.

[15]

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

[16]

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

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.

[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.

[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.

[4]

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

[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.

[6]

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

[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.

[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.

[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.

[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.

[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.

[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.

[13]

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

[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.

[15]

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

[16]

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

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