Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
권호 표지
DOI QR코드

DOI QR Code

QUADRATIC RESIDUE CODES OVER GALOIS RINGS

  • Received : 2016.08.09
  • Accepted : 2016.09.19
  • Published : 2016.09.30
Download PDF
⟨ Previous Next ⟩

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.

Keywords

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.