Abstract
We consider ways in which multipoint algebraic geometry codes may be viewed as subcodes of the more traditionally studied one-point codes. Examples are provided to illustrate the impact of choices made on this embedding.
This is a preview of subscription content, log in via an institution to check access.
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
P. Beelen, The order bound for general algebraic geometric codes, Finite Fields Appl. 13 (2007), no. 3, 655–680.
C. Carvalho and F. Torres, On Goppa codes and Weierstrass gaps at several points, Des. Codes Cryptogr. 35 (2005), no. 2, 211–225.
G. L. Feng and T. R. N. Rao, Decoding algebraic-geometric codes up to the designed minimum distance, IEEE Trans. on Inf. Th. 39 (1993), no. 1, 37–45.
G. L. Feng and T. R. N. Rao, A simple approach for construction of algebraic-geometric codes from affine plane curves, IEEE Trans. on Inf. Th. 40 (1994), 1003–1012.
G. L. Feng and T. R. N. Rao, Improved geometric Goppa codes, Part I: Basic theory, IEEE Trans. on Inf. Th. 41 (1995), 1678–1693.
V. D. Goppa, Codes on algebraic curves, Soviet Math. Dokl. 24 (1981), no. 1, 170–172.
M. Homma and S. J. Kim, Goppa codes with Weierstrass pairs, J. Pure Appl. Algebra 162 (2001), nos. 2–3, 273–290.
M. Homma and S. J. Kim, Toward the determination of the minimum distance of two-point codes on a Hermitian curve, Des. Codes Cryptogr. 37 (2005), no. 1, 111–132.
M. Homma and S. J. Kim, The two-point codes on a Hermitian curve with the designed minimum distance, Des. Codes Cryptogr. 38 (2006a), no. 1, 55–81.
M. Homma and S. J. Kim, The two-point codes with the designed distance on a Hermitian curve in even characteristic, Des. Codes Crypto. 39 (2006b), no. 3, 375–386.
T. Høholdt, J. van Lint, and R. Pellikaan, Algebraic geometry of codes, Handbook of Coding Theory (V. S. Pless and W.C. Huffman, eds.), Elsevier, Amsterdam, 1998, pp. 871–961.
D. A. Leonard, A tutorial on AG code construction from a Gröbner basis perspective, this volume, 2009, pp. 93–106.
G. L. Matthews, Weierstrass pairs and minimum distance of Goppa codes, Des. Codes Cryptogr. 22 (2001), no. 2, 107–121.
H. Maharaj, G. L. Matthews, and G. Pirsic, Riemann-Roch spaces of the Hermitian function field with applications to algebraic geometry codes and low-discrepancy sequences, J. Pure Appl. Algebra 195 (2005), no. 3, 261–280.
H. G. Rück and H. Stichtenoth, A characterization of Hermitian function fields over finite fields, J. Reine Angew. Math. 457 (1994), 185–188.
H. Stichtenoth, Algebraic function fields and codes, Universitext, Springer, Berlin, 1993.
M. A. Tsfasman and S. G. Vlăduţ, Algebraic-geometric codes, Math. and its Appl. (Soviet Series), vol. 58, Kluwer Academic, Dordrecht, 1991.
K. Yang and P. V. Kumar, On the true minimum distance of Hermitian codes, Proc. of AGCT 1991, LNM, vol. 1518, Springer, Berlin, 1992, pp. 99–107.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Matthews, G.L. (2009). Viewing Multipoint Codes as Subcodes of One-Point Codes. In: Sala, M., Sakata, S., Mora, T., Traverso, C., Perret, L. (eds) Gröbner Bases, Coding, and Cryptography. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-93806-4_28
Download citation
DOI: https://doi.org/10.1007/978-3-540-93806-4_28
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-93805-7
Online ISBN: 978-3-540-93806-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)