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On Constructing AG Codes Without Basis Functions for Riemann-Roch Spaces

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Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3857))

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Abstract

The standard construction of linear error-correcting codes on algebraic curves requires determining a basis for the Riemann-Roch space \(\mathcal{L}\)(G)  associated to a given divisor G, often a hard problem. Here we consider the problem of constructing the code without any knowledge of such a basis. We interpret the columns of a generator matrix as points on an embedded copy of the curve, and show that in certain cases these points can be realized in principle as the images of a set of vector bundles under a standard map to a class of repartitions.

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References

  1. Atiyah, M.: Complex fibre bundles and ruled surfaces. Proc. London Math. Soc. 5, 407–434 (1955)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bertram, A.: Moduli of rank-2 vector bundles, theta divisors, and the geometry of curves in projective space. Journal of Diff. Geometry 35, 429–469 (1992)

    MATH  MathSciNet  Google Scholar 

  3. Bogomolov, F., Petrov, T.: Algebraic Curves and One-Dimensional Fields. Courant Lecture Notes in Mathematics. Amer. Math. Soc., Providence (2002)

    MATH  Google Scholar 

  4. Coles, D.: Vector Bundles and Codes on the Hermitian Curve. IEEE Trans. Inf. Theory 51(6), 2113–2120 (2005)

    Article  MathSciNet  Google Scholar 

  5. Goppa, V.D.: Codes on algebraic curves. Soviet Math. Dokl. 24, 170–172 (1981)

    MATH  Google Scholar 

  6. Garcia, A., Stichtenoth, H.: A Tower of Artin-Schreier extensions of function fields attaining the Drinfeld-Vlǎduţ bound. Invent. Math. 121, 211–222 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  7. Hartshorne, R.: Algebraic Geometry. Springer, New York (1977)

    MATH  Google Scholar 

  8. Høholdt, T., Pellikaan, R.: On the decoding of algebraic-geometric codes. IEEE Trans. Inf. Theory 41(6), 1589–1614 (1995)

    Article  Google Scholar 

  9. Johnsen, T.: Rank two bundles on Algebraic Curves and decoding of Goppa Codes. International Journal of Pure and Applied Mathematics 4(1), 33–45 (2003)

    MATH  MathSciNet  Google Scholar 

  10. Lange, H., Narasimhan, M.S.: Maximal Subbundles of Rank Two Vector Bundles on Curves. Math. Ann. 266, 55–72 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  11. Pretzel, O.: Codes and Algebraic Curves. Oxford University Press, Oxford (1998)

    MATH  Google Scholar 

  12. Shararevich, I.R.: Basic Algebraic Geometry 1-2, 2nd edn. Springer, New York (1994); Translated by Reid, M

    Google Scholar 

  13. Shum, K.W., Aleshnikov, I., Kumar, P.V., Stichtenoth, H., Deolalikar, V.: A Low-Complexity Algorithm for the Construction of Algebraic-Geometric Codes Better Than the Gilbert-Varshamov Bound. IEEE Trans. Inf. Theory 47(6), 2225–2241 (2001)

    Article  MATH  Google Scholar 

  14. Serre, J.-P.: Algebraic Groups and Class Fields. Springer, New York (1988)

    MATH  Google Scholar 

  15. Tsfasman, M.A., Vlǎduţ, S.G., Zink, T.: Modular curves, Shimura curves and Goppa codes better than the Varshamov-Gilbert bound. Mathematische Nachrichten 109, 21–28 (1982)

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Computer Science, and Statistics, Bloomsburg University,  

    Drue Coles

Authors
  1. Drue Coles
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Editor information

Editors and Affiliations

  1. Department of Electrical Engineering, University of Hawaii, Honolulu, USA

    Marc P. C. Fossorier

  2. National Institute of Advanced Industrial Science and Technology (AIST), Research Center for Information Security (RCIS),  

    Hideki Imai

  3. Department of Electrical and Computer Engineering, University of California, CA 95616, Davis,  

    Shu Lin

  4. University Paul Sabatier, AAECC/IRIT, 118 route de Narbonne, 31300, Toulouse cedex, France

    Alain Poli

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© 2006 Springer-Verlag Berlin Heidelberg

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Coles, D. (2006). On Constructing AG Codes Without Basis Functions for Riemann-Roch Spaces. In: Fossorier, M.P.C., Imai, H., Lin, S., Poli, A. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 2006. Lecture Notes in Computer Science, vol 3857. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11617983_10

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  • DOI: https://doi.org/10.1007/11617983_10

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-31423-3

  • Online ISBN: 978-3-540-31424-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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