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

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.

Keywords

• AG codes,Gröbner basis,Root diagram

References

1. [1] W. Adams, P. Loustaunau, An Introduction to Gröbner Bases, Providence, RI: American Mathematical Society, 1994.
2. [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. [3] D. Cox, J. Little, D. O’Shea, Using Algebraic Geometry, Springer, New York, 1998.
4. [4] J. I. Farrán, C. Munuera, G. Tizziotti, F. Torres, Gröbner basis for norm–trace codes, J. Symb. Comput. 48 (2013) 54–63.
5. [5] A. Garcia, P. Viana, Weierstrass points on certain non–classical curves, Arch. Math. 46(4) (1986) 315–322.
6. [6] V. D. Goppa, Codes on algebraic curves, Dokl. Akad. Nauk SSSR 259(6) (1981) 1289–1290.
7. [7] V. D. Goppa, Algebraic–geometric codes, Izv. Akad. Nauk SSSR Ser. Mat. 46(4) (1982) 762–781.
8. [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. [9] J. W. P. Hirschfeld, G. Korchmáros, F. Torres, Algebraic Curves over a Finite Field, Princeton University Press, Princeton, 2008.
10. [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. [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. [12] J. Little, K. Saints, C. Heegard, On the structure of Hermitian codes, J. Pure Appl. Algebra, 121(3) (1997) 293–314.
13. [13] G. L. Matthews, Weierstrass semigroups and codes from a quotient of the Hermitian curve, Des. Codes Cryptogr. 37(3) (2005) 473–492.
14. [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. [15] H. Stichtenoth, Algebraic Function Fields and Codes, Springer–Verlag, Berlin, 1993.