Abstract
In this paper, we consider the norm-trace curves which are defined by the equation \(y^{q^{r-1}}+y^{q^{r-2}}+ \cdots +y=x^{\frac{q^r-1}{q-1}}\) over where q is a power of a prime number and r ≥ 2 is an integer. We determine the Weierstrass semigroup of the triple of points \(\left(P_{\infty}, P_{00}, P_{0b} \right)\) on this curve.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Arbarello, E., Cornalba, M., Griffiths, P., Harris, J.: Geometry of Algebraic Curves. Springer, Heidelberg (1985)
Ballico, E., Kim, S.J.: Weierstrass multiple loci of n-pointed algebraic curves. J. Algebra 199(2), 455–471 (1998)
Carvalho, C., Kato, T.: On Weierstrass semigroups and sets: a review of new results, Geom. Dedicata (to appear)
Carvalho, C., Torres, F.: On Goppa codes and Weierstrass gaps at several points. Des. Codes Cryptogr. 35(2), 211–225 (2005)
GarcÃa, A., Kim, S.J., Lax, R.F.: Consecutive Weierstrass gaps and minimum distance of Goppa codes. J. Pure Appl. Algebra 84(2), 199–207 (1993)
Geil, O.: On codes from norm-trace curves. Finite Fields Appl. 9(3), 351–371 (2003)
Homma, M., Kim, S.J.: Goppa codes with Weierstrass pairs. J. Pure Appl. Algebra 162(2-3), 273–290 (2001)
Ishii, N.: A certain graph obtained from a set of several points on a Riemann surface. Tsukuba J. Math. 23(1), 55–89 (1999)
Kim, S.J.: On the index of the Weierstrass semigroup of a pair of points on a curve. Arch. Math (Basel) 62(1), 73–82 (1994)
Matthews, G.L.: Codes from the Suzuki function field. IEEE Trans. Inform. Theory 50(12), 3298–3302 (2004)
Matthews, G.L.: Some computational tools for estimating the parameters of algebraic geometry codes. In: Coding theory and quantum computing, Contemp. Math., vol. 381, pp. 19–26. Amer. Math. Soc., Providence (2005)
Matthews, G.L.: Weierstrass semigroups and codes from a quotient of the Hermitian curve. Des. Codes Cryptogr. 37(3), 473–492 (2005)
Matthews, G.L.: The Weierstrass semigroup of an m-tuple of collinear points on a Hermitian curve. In: Mullen, G.L., Poli, A., Stichtenoth, H. (eds.) Fq7 2003. LNCS, vol. 2948, pp. 12–24. Springer, Heidelberg (2004)
Matthews, G.L.: Weierstrass pairs and minimum distance of Goppa codes. Des. Codes Cryptogr. 22(2), 107–121 (2001)
Munuera, C., Tizziotti, G.C., Torres, F.: Two-point codes on Norm-Trace curves. In: Barbero, A. (ed.) ICMCTA 2008. LNCS, vol. 5228, pp. 128–136. Springer, Heidelberg (2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Matthews, G.L. (2009). On Weierstrass Semigroups of Some Triples on Norm-Trace Curves. In: Chee, Y.M., Li, C., Ling, S., Wang, H., Xing, C. (eds) Coding and Cryptology. IWCC 2009. Lecture Notes in Computer Science, vol 5557. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01877-0_13
Download citation
DOI: https://doi.org/10.1007/978-3-642-01877-0_13
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-01813-8
Online ISBN: 978-3-642-01877-0
eBook Packages: Computer ScienceComputer Science (R0)