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A note on cohomology and algebraic geometric codes on the curves over rings Cover

A note on cohomology and algebraic geometric codes on the curves over rings

Open Access
|Jul 2025

References

  1. K. G. Bartley, J. L. Walker, Algebraic geometric codes over rings, Advances in Algebraic Geometry Codes, Series on Coding Theory and Cryptology, Vol. 5, World Scientific Publishing Co. Pte. Ltd., 2008.
  2. R. Blache, L-functions of exponential sums on curves over rings, Finite Fields Appl. 15 (2009), 345-359.
  3. R. Cramer, M. Rambaud, C. Xing, Asymptotically-good arithmetic secret sharing over/plwith strong multiplication and its applications to efficient MPC, Cryptology ePrint Archive, Paper 2019/832, 2019.
  4. D. Eisenbud, Commutative Algebra, with a view toward algebraic geometry, Graduate Texts in Mathematics, Springer, 1995.
  5. A. Gathmann, Algebraic geometry, Class notes, TU Kaiserslautern, 2014.
  6. V. D. Goppa, Codes associated with divisors, Probl. Peredachi Inf. 13 (1977), 33-39, English translation in Probl. Inf. Transm. 13 (1977), 22-27.
  7. A. Grothendieck,Éléments de géométrie algébrique: I. Le langage des schémas, Publications Mathématiques de l’IHÉS Vol. 4, 1960, 5-228.
  8. R. Hartshorne, Residues and duality, Springer-Verlag, New York, 1966.
  9. R. Hartshorne, Algebraic geometry, Springer-Verlag, New York, 1977.
  10. A. J. de Jong et al, Stacks Project, Version 2dce121a, compiled on April 12, 2021, https://stacks.math.columbia.edu.
  11. Q. Liu, Algebraic geometry and arithmetic curves, Oxford University Press 2002.
  12. R. Pellikaan, B. - Z. Shen, G. J. M. Van Wee, Which linear codes are algebraic-geometric?, IEEE Trans. Inf. Theory 37 (1991), 583-602.
  13. C. Peskine, Introduction algébrique à la géométrie projective, Analyse complexe et géométrie, Université Paris 6, 2007.
  14. M. A. Tsfasman, S. G. Vlăduţ, Th. Zink, Modular curves, Shimura curves, and Goppa codes, better than the Varshamov-Gilbert bound, Math. Nachr. ℤ (1982), 21-28.
  15. J. F. Voloch, J. L. Walker, Lee weights of/4ℤ-codes from elliptic curves, Codes, Curves, and Signals: Common Threads in Communications (1998), 53-62.
  16. J. F. Voloch, J. L. Walker, Euclidean weights of codes from elliptic curves over rings, Trans. Am. Math. Soc. 352 (11) (2000), 5063-5076.
  17. J. F. Voloch, J. L. Walker, Codes over rings from curves of higher genus, IEEE Trans. Inf. Theory 45 (1999), 1768-1776.
  18. J. L. Walker, Algebraic geometric codes over rings, PhD thesis, University of Illinois at Urbana-Champaign, 1996.
  19. J. L. Walker, The Nordstrom-Robinson code is algebraic geometric, IEEE Trans. Inf. Theory 43 (5) (1997), 1588-1593.
  20. J. L. Walker, Algebraic geometric codes over rings, J. Pure Appl. Algebra 144 (1) (1999), 91-110.
DOI: https://doi.org/10.2478/awutm-2025-0007 | Journal eISSN: 1841-3307 | Journal ISSN: 1841-3293
Language: English
Page range: 75 - 88
Submitted on: Feb 16, 2025
Accepted on: Jun 16, 2025
Published on: Jul 5, 2025
Published by: West University of Timisoara
In partnership with: Paradigm Publishing Services
Publication frequency: Volume open

© 2025 Francis Nyamda, Christophe Mouaha, published by West University of Timisoara
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.