2014, 8(1): 67-72. doi: 10.3934/amc.2014.8.67

Weierstrass semigroup and codes over the curve $y^q + y = x^{q^r + 1}$

1. 

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

Received  November 2012 Published  January 2014

We compute the Weierstrass semigroup at a pair of rational points on the curve defined by the affine equation $y^q + y = x^{q^r + 1}$ over $\mathbb{F}_{q^{2r}}$, where $r$ is a positive odd integer and $q$ is a prime power. We then construct a two-point AG code on the curve whose relative parameters are better than comparable one-point AG code.
Citation: Alonso Sepúlveda, Guilherme Tizziotti. Weierstrass semigroup and codes over the curve $y^q + y = x^{q^r + 1}$. Advances in Mathematics of Communications, 2014, 8 (1) : 67-72. doi: 10.3934/amc.2014.8.67
References:
[1]

E. Arbarello, M. Cornalba, P. Griffiths and J. Harris, Geometry of Algebraic Curves,, Springer-Verlag, (1985).

[2]

E. Ballico, Weierstrass points and Weierstrass pairs on algebraic curves,, Int. J. Pure Appl. Math., 2 (2002), 427.

[3]

C. Carvalho and T. Kato, On Weierstrass semigroup and sets: a review with new results,, Geom. Dedicata, 139 (2009), 139. doi: 10.1007/s10711-008-9337-y.

[4]

I. M. Duursma, R. Kirov, Improved two-point codes on Hermitian curves,, IEEE Trans. Inf. Theory, 57 (2011), 4469. doi: 10.1109/TIT.2011.2146410.

[5]

T. Hasegawa, S. Kondo and H. Kurusu, A sequence of one-point codes from a tower of function fields,, Des. Codes Crypt., 41 (2006), 251. doi: 10.1007/s10623-006-9013-x.

[6]

T. Høholdt, J. van Lint and R. Pellikaan, Algebraic Geometry Codes,, Elsevier, (1998).

[7]

M. Homma, The Weierstrass semigroup of a pair of points on a curve,, Arch. Math., 67 (1996), 337. doi: 10.1007/BF01197599.

[8]

S. J. Kim, On index of the Weierstrass semigroup of a pair of points on a curve,, Arch. Math., 62 (1994), 73. doi: 10.1007/BF01200442.

[9]

S. Kondo, T. Katagiri and T. Ogihara, Automorphism groups of one-point codes from the curves $y^q + y = x^{q^r+1}$,, IEEE Trans. Inf. Theory, 47 (2001), 2573. doi: 10.1109/18.945272.

[10]

G. L. Matthews, Weierstrass pairs and minimum distance of Goppa codes,, Des. Codes Crypt., 22 (2001), 107. doi: 10.1023/A:1008311518095.

[11]

G. L. Matthews, Codes from the Suzuki function field,, IEEE Trans. Inf. Theory, 50 (2004), 3298. doi: 10.1109/TIT.2004.838102.

[12]

C. Munuera, A. Sepulveda and F. Torres, Castle curve and codes,, Adv. Math. Commun., 3 (2009), 399. doi: 10.3934/amc.2009.3.399.

[13]

H. Stichtenoth, Algebraic Function Fields and Codes,, Springer, (1993).

[14]

M. Tsfasman, S. Vlădut and D. Nogin, Algebraic Geometric Codes: Basic Notions,, Amer. Math. Soc., (2007).

[15]

J. H. van Lint, Introduction to Coding Theory,, Springer, (1982).

show all references

References:
[1]

E. Arbarello, M. Cornalba, P. Griffiths and J. Harris, Geometry of Algebraic Curves,, Springer-Verlag, (1985).

[2]

E. Ballico, Weierstrass points and Weierstrass pairs on algebraic curves,, Int. J. Pure Appl. Math., 2 (2002), 427.

[3]

C. Carvalho and T. Kato, On Weierstrass semigroup and sets: a review with new results,, Geom. Dedicata, 139 (2009), 139. doi: 10.1007/s10711-008-9337-y.

[4]

I. M. Duursma, R. Kirov, Improved two-point codes on Hermitian curves,, IEEE Trans. Inf. Theory, 57 (2011), 4469. doi: 10.1109/TIT.2011.2146410.

[5]

T. Hasegawa, S. Kondo and H. Kurusu, A sequence of one-point codes from a tower of function fields,, Des. Codes Crypt., 41 (2006), 251. doi: 10.1007/s10623-006-9013-x.

[6]

T. Høholdt, J. van Lint and R. Pellikaan, Algebraic Geometry Codes,, Elsevier, (1998).

[7]

M. Homma, The Weierstrass semigroup of a pair of points on a curve,, Arch. Math., 67 (1996), 337. doi: 10.1007/BF01197599.

[8]

S. J. Kim, On index of the Weierstrass semigroup of a pair of points on a curve,, Arch. Math., 62 (1994), 73. doi: 10.1007/BF01200442.

[9]

S. Kondo, T. Katagiri and T. Ogihara, Automorphism groups of one-point codes from the curves $y^q + y = x^{q^r+1}$,, IEEE Trans. Inf. Theory, 47 (2001), 2573. doi: 10.1109/18.945272.

[10]

G. L. Matthews, Weierstrass pairs and minimum distance of Goppa codes,, Des. Codes Crypt., 22 (2001), 107. doi: 10.1023/A:1008311518095.

[11]

G. L. Matthews, Codes from the Suzuki function field,, IEEE Trans. Inf. Theory, 50 (2004), 3298. doi: 10.1109/TIT.2004.838102.

[12]

C. Munuera, A. Sepulveda and F. Torres, Castle curve and codes,, Adv. Math. Commun., 3 (2009), 399. doi: 10.3934/amc.2009.3.399.

[13]

H. Stichtenoth, Algebraic Function Fields and Codes,, Springer, (1993).

[14]

M. Tsfasman, S. Vlădut and D. Nogin, Algebraic Geometric Codes: Basic Notions,, Amer. Math. Soc., (2007).

[15]

J. H. van Lint, Introduction to Coding Theory,, Springer, (1982).

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