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Year 2018, Volume: 5 Issue: 2, 71 - 83, 15.05.2018
https://doi.org/10.13069/jacodesmath.423733

Abstract

References

  • [1] W. Adams, P. Loustaunau, An Introduction to Gröbner Bases, Providence, RI: American Mathematical Society, 1994.
  • [2] A. S. Castellanos, A. M. Masuda, L. Quoos, One– and two–point codes over Kummer extensions, IEEE Trans. Inform. Theory 62(9) (2016) 4867–4872.
  • [3] D. Cox, J. Little, D. O’Shea, Using Algebraic Geometry, Springer, New York, 1998.
  • [4] J. I. Farrán, C. Munuera, G. Tizziotti, F. Torres, Gröbner basis for norm–trace codes, J. Symb. Comput. 48 (2013) 54–63.
  • [5] A. Garcia, P. Viana, Weierstrass points on certain non–classical curves, Arch. Math. 46(4) (1986) 315–322.
  • [6] V. D. Goppa, Codes on algebraic curves, Dokl. Akad. Nauk SSSR 259(6) (1981) 1289–1290.
  • [7] V. D. Goppa, Algebraic–geometric codes, Izv. Akad. Nauk SSSR Ser. Mat. 46(4) (1982) 762–781.
  • [8] C. Heegard, J. Little, K. Saints, Systematic encoding via Gröbner bases for a class of algebraic–geometric Goppa codes, IEEE Trans. Inform. Theory 41(6) (1995) 1752–1761.
  • [9] J. W. P. Hirschfeld, G. Korchmáros, F. Torres, Algebraic Curves over a Finite Field, Princeton University Press, Princeton, 2008.
  • [10] T. Høholdt, J. van Lint, R. Pellikaan, Algebraic geometry codes, in Handbook of Coding Theory, V. S. Pless, W. C. Huffman, R. A. Brualdi (Eds.), v. 1, Elsevier, Amsterdam, 1998, 871–961.
  • [11] S. Kondo, T. Katagiri, T. Ogihara, Automorphism groups of one–point codes from the curves $y^q + y = x{q^r+1}$, IEEE Trans. Inform. Theory 47(6) (2001) 2573–2579.
  • [12] J. Little, K. Saints, C. Heegard, On the structure of Hermitian codes, J. Pure Appl. Algebra, 121(3) (1997) 293–314.
  • [13] G. L. Matthews, Weierstrass semigroups and codes from a quotient of the Hermitian curve, Des. Codes Cryptogr. 37(3) (2005) 473–492.
  • [14] A. Sepúlveda, G. Tizziotti, Weierstrass semigroup and codes over the curve $y^q + y = x{q^r+1}$, Adv. Math. Commun. 8(1) (2014) 67–72.
  • [15] H. Stichtenoth, Algebraic Function Fields and Codes, Springer–Verlag, Berlin, 1993.

The root diagram for one-point AG codes arising from certain curves with separated variables

Year 2018, Volume: 5 Issue: 2, 71 - 83, 15.05.2018
https://doi.org/10.13069/jacodesmath.423733

Abstract

Heegard, Little and Saints introduced in [8] an encoding algorithm for a class of AG codes via Gröbner
basis more compact compared with the usual encoding via generator matrix. So, knowing that the
main drawback of Gröbner basis is the high computational cost required for its calculation, in [12],
the same authors introduced the concept of root diagram that allows the construction of an algorithm
for computing a Gröbner basis with a lower complexity for one-point Hermitian codes. In [4], Farrán,
Munuera, Tizziotti and Torres extended the results obtained in [12] for codes on norm-trace curves.
In this work we generalize these results by constructing the root diagram for codes arising from certain
curves with separated variables that has certain special automorphism and a Weierstrass semigroup
generated by two elements. Such family of curves includes the norm-trace curve, among other curves
with recent applications in coding theory.

References

  • [1] W. Adams, P. Loustaunau, An Introduction to Gröbner Bases, Providence, RI: American Mathematical Society, 1994.
  • [2] A. S. Castellanos, A. M. Masuda, L. Quoos, One– and two–point codes over Kummer extensions, IEEE Trans. Inform. Theory 62(9) (2016) 4867–4872.
  • [3] D. Cox, J. Little, D. O’Shea, Using Algebraic Geometry, Springer, New York, 1998.
  • [4] J. I. Farrán, C. Munuera, G. Tizziotti, F. Torres, Gröbner basis for norm–trace codes, J. Symb. Comput. 48 (2013) 54–63.
  • [5] A. Garcia, P. Viana, Weierstrass points on certain non–classical curves, Arch. Math. 46(4) (1986) 315–322.
  • [6] V. D. Goppa, Codes on algebraic curves, Dokl. Akad. Nauk SSSR 259(6) (1981) 1289–1290.
  • [7] V. D. Goppa, Algebraic–geometric codes, Izv. Akad. Nauk SSSR Ser. Mat. 46(4) (1982) 762–781.
  • [8] C. Heegard, J. Little, K. Saints, Systematic encoding via Gröbner bases for a class of algebraic–geometric Goppa codes, IEEE Trans. Inform. Theory 41(6) (1995) 1752–1761.
  • [9] J. W. P. Hirschfeld, G. Korchmáros, F. Torres, Algebraic Curves over a Finite Field, Princeton University Press, Princeton, 2008.
  • [10] T. Høholdt, J. van Lint, R. Pellikaan, Algebraic geometry codes, in Handbook of Coding Theory, V. S. Pless, W. C. Huffman, R. A. Brualdi (Eds.), v. 1, Elsevier, Amsterdam, 1998, 871–961.
  • [11] S. Kondo, T. Katagiri, T. Ogihara, Automorphism groups of one–point codes from the curves $y^q + y = x{q^r+1}$, IEEE Trans. Inform. Theory 47(6) (2001) 2573–2579.
  • [12] J. Little, K. Saints, C. Heegard, On the structure of Hermitian codes, J. Pure Appl. Algebra, 121(3) (1997) 293–314.
  • [13] G. L. Matthews, Weierstrass semigroups and codes from a quotient of the Hermitian curve, Des. Codes Cryptogr. 37(3) (2005) 473–492.
  • [14] A. Sepúlveda, G. Tizziotti, Weierstrass semigroup and codes over the curve $y^q + y = x{q^r+1}$, Adv. Math. Commun. 8(1) (2014) 67–72.
  • [15] H. Stichtenoth, Algebraic Function Fields and Codes, Springer–Verlag, Berlin, 1993.
There are 15 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Federico Fornasiero This is me

Guilherme Tizziotti This is me 0000-0003-1026-0546

Publication Date May 15, 2018
Published in Issue Year 2018 Volume: 5 Issue: 2

Cite

APA Fornasiero, F., & Tizziotti, G. (2018). The root diagram for one-point AG codes arising from certain curves with separated variables. Journal of Algebra Combinatorics Discrete Structures and Applications, 5(2), 71-83. https://doi.org/10.13069/jacodesmath.423733
AMA Fornasiero F, Tizziotti G. The root diagram for one-point AG codes arising from certain curves with separated variables. Journal of Algebra Combinatorics Discrete Structures and Applications. May 2018;5(2):71-83. doi:10.13069/jacodesmath.423733
Chicago Fornasiero, Federico, and Guilherme Tizziotti. “The Root Diagram for One-Point AG Codes Arising from Certain Curves With Separated Variables”. Journal of Algebra Combinatorics Discrete Structures and Applications 5, no. 2 (May 2018): 71-83. https://doi.org/10.13069/jacodesmath.423733.
EndNote Fornasiero F, Tizziotti G (May 1, 2018) The root diagram for one-point AG codes arising from certain curves with separated variables. Journal of Algebra Combinatorics Discrete Structures and Applications 5 2 71–83.
IEEE F. Fornasiero and G. Tizziotti, “The root diagram for one-point AG codes arising from certain curves with separated variables”, Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 5, no. 2, pp. 71–83, 2018, doi: 10.13069/jacodesmath.423733.
ISNAD Fornasiero, Federico - Tizziotti, Guilherme. “The Root Diagram for One-Point AG Codes Arising from Certain Curves With Separated Variables”. Journal of Algebra Combinatorics Discrete Structures and Applications 5/2 (May 2018), 71-83. https://doi.org/10.13069/jacodesmath.423733.
JAMA Fornasiero F, Tizziotti G. The root diagram for one-point AG codes arising from certain curves with separated variables. Journal of Algebra Combinatorics Discrete Structures and Applications. 2018;5:71–83.
MLA Fornasiero, Federico and Guilherme Tizziotti. “The Root Diagram for One-Point AG Codes Arising from Certain Curves With Separated Variables”. Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 5, no. 2, 2018, pp. 71-83, doi:10.13069/jacodesmath.423733.
Vancouver Fornasiero F, Tizziotti G. The root diagram for one-point AG codes arising from certain curves with separated variables. Journal of Algebra Combinatorics Discrete Structures and Applications. 2018;5(2):71-83.