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QUADRATIC RESIDUE CODES OVER GALOIS RINGS

  • Park, Young Ho (Department of Mathematics Kangwon National University)
  • Received : 2016.08.09
  • Accepted : 2016.09.19
  • Published : 2016.09.30

Abstract

Quadratic residue codes are cyclic codes of prime length n defined over a finite field ${\mathbb{F}}_{p^e}$, where $p^e$ is a quadratic residue mod n. They comprise a very important family of codes. In this article we introduce the generalization of quadratic residue codes defined over Galois rings using the Galois theory.

Acknowledgement

Supported by : Kangwon National University

References

  1. A.R.Calderbank and N.J.A. Sloane, Modular and p-adic cyclic codes, Des. Codes. Cryptogr. 6 (1995), 21-35. https://doi.org/10.1007/BF01390768
  2. M.H. Chiu, S.S.Yau and Y. Yu, ${\mathbb{Z}}_8$-cyclic codes and quadratic residue codes, Advances in Applied Math. 25 (2000), 12-33. https://doi.org/10.1006/aama.2000.0687
  3. S.T. Dougherty, S.Y. Kim and Y.H. Park, Lifted codes and their weight enumerators, Discrite Math. 305 (2005), 123-135. https://doi.org/10.1016/j.disc.2005.08.004
  4. W.C. Huffman and V. Pless, Fundamentals of error-correcting codes, Cambridge, 2003.
  5. S. J. Kim, Quadratic residue codes over ${\mathbb{Z}}_{16}$, Kangweon-Kyungki Math. J. 11 (2003), 57-64.
  6. S. J. Kim, Generator polynomials of the p-adic quadratic residue codes, Kangweon-Kyungki Math. J. 13 (2005), 103-112.
  7. B. McDonald, Finite rings with identity, Marcel Dekker, 1974.
  8. Y.H. Park, Quadratic residue codes over p-adic integers and their projections to integers modulo $p^e$, Korean J. Math. 23 (2015), 163-169. https://doi.org/10.11568/kjm.2015.23.1.163
  9. V.S. Pless and Z. Qian, Cyclic codes and quadratic residue codes over ${\mathbb{Z}}_4$, IEEE Trans. Inform. Theory. 42 (1996), 1594-1600. https://doi.org/10.1109/18.532906
  10. B. Taeri, Quadratic residue codes over ${\mathbb{Z}}_9$, J. Korean Math Soc. 46 (2009), 13-30. https://doi.org/10.4134/JKMS.2009.46.1.013
  11. X. Tan, A family of quadratic residue codes over ${\mathbb{Z}}_{2m}$, preprint, 2011.