DOI QR코드

DOI QR Code

FAST UNIQUE DECODING OF PLANE AG CODES

  • Received : 2013.11.07
  • Accepted : 2013.11.25
  • Published : 2013.12.25

Abstract

An interpolation-based unique decoding algorithm of Algebraic Geometry codes was recently introduced. The algorithm iteratively computes the sent message through a majority voting procedure using the Gr$\ddot{o}$bner bases of interpolation modules. We now combine the main idea of the Guruswami-Sudan list decoding with the algorithm, and thus obtain a hybrid unique decoding algorithm of plane AG codes, significantly improving the decoding speed.

Keywords

algebraic geometry codes;interpolation decoding;Gr$\ddot{o}$bner bases

References

  1. M. F. Atiyah and I. G. MacDonald, Introduction to Commutative Algebra, Perseus Books, 1969.
  2. D. Eisenbud, Commutative Algebra with a View toward Algebraic Geometry, Number 150 in GTM, Springer-Verlag, 1995.
  3. K. Lee, Unique decoding of plane AG codes revisited, arXiv, Apr. 2012. arXiv:1204.0052v2.
  4. K. Lee, M. Bras-Amoros, and M. E. O'Sullivan, Unique decoding of plane AG codes via interpolation, IEEE Trans. Inf. Theory, 58(6) (2012), 3941-3950. https://doi.org/10.1109/TIT.2012.2182757
  5. K. Lee and M. E. O'Sullivan, List decoding of Hermitian codes using Grobner bases, Journal of Symbolic Computation, 44 (2009), 1662-1675. https://doi.org/10.1016/j.jsc.2007.12.004
  6. J. C. Rosales and P. A. Garca-Sanchez, Numerical semigroups, volume 20 of Developments in Mathematics, Springer, New York, 2009.
  7. R. M. Roth and G. Ruckenstein, Ecient decoding of Reed-Solomon codes beyond half the minimum distance, IEEE Trans. Inf. Theory, 46(1) (2000), 246-257. https://doi.org/10.1109/18.817522
  8. W. A. Stein et al, Sage Mathematics Software (Version 4.8), The Sage Development Team, 2012. http://www.sagemath.org

Acknowledgement

Supported by : National Research Foundation of Korea (NRF)