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

Alonso Sepúlveda; Guilherme Tizziotti

doi:10.3934/amc.2014.8.67

Article Contents

2014, Volume 8, Issue 1: 67-72. Doi: 10.3934/amc.2014.8.67

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

Alonso Sepúlveda^1, and
Guilherme Tizziotti^1,

1.
Universidade Federal de Uberlandia, Campus Santa Monica, Av. Joao Naves de Avila, 2121, Uberlandia-MG, CEP 38.408-100

Received: October 31, 2012

Published: January 2014

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Abstract

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.

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Mathematics Subject Classification: Primary: 14H55; Secondary: 11G20, 14G50.

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References

[1]	E. Arbarello, M. Cornalba, P. Griffiths and J. Harris, Geometry of Algebraic Curves, Springer-Verlag, Berlin, 1985.
[2]	E. Ballico, Weierstrass points and Weierstrass pairs on algebraic curves, Int. J. Pure Appl. Math., 2 (2002), 427-440.
[3]	C. Carvalho and T. Kato, On Weierstrass semigroup and sets: a review with new results, Geom. Dedicata, 139 (2009), 139-195.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(7) (2011), 4469-4476.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), 251-267.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), 337-348.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), 73-82.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), 2573-2579.doi: 10.1109/18.945272.
[10]	G. L. Matthews, Weierstrass pairs and minimum distance of Goppa codes, Des. Codes Crypt., 22 (2001), 107-121.doi: 10.1023/A:1008311518095.
[11]	G. L. Matthews, Codes from the Suzuki function field, IEEE Trans. Inf. Theory, 50(12) (2004), 3298-3302.doi: 10.1109/TIT.2004.838102.
[12]	C. Munuera, A. Sepulveda and F. Torres, Castle curve and codes, Adv. Math. Commun., 3 (2009), 399-408.doi: 10.3934/amc.2009.3.399.
[13]	H. Stichtenoth, Algebraic Function Fields and Codes, Springer, Berlin, 1993.
[14]	M. Tsfasman, S. Vlădut and D. Nogin, Algebraic Geometric Codes: Basic Notions, Amer. Math. Soc., Providence, 2007.
[15]	J. H. van Lint, Introduction to Coding Theory, Springer, New York, 1982.