DOI QR코드

DOI QR Code

IDEALS OF Zpn[X]/(Xl-1)

  • Woo, Sung-Sik (Department of Mathematics Ewha Women's University)
  • Received : 2010.04.06
  • Published : 2011.07.31

Abstract

In [6, 8], we showed that any ideal of $\mathbb{Z}_4[X]/(X^l\;-\;1)$ is generated by at most two polynomials of the `standard' forms when l is even. The purpose of this paper is to find the `standard' generators of the cyclic codes over $\mathbb{Z}_{p^a}$ of length a multiple of p, namely the ideals of $\mathbb{Z}_{p^a}[X]/(X^l\;-\;1)$ with an integer l which is a multiple of p. We also find an explicit description of their duals in terms of the generators when a = 2.

Keywords

cyclic code over $\mathbb{Z}_{p^a}$;ideals of $\mathbb{Z}_{p^n}[X]/(X^l\-\1)$

References

  1. M. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1969.
  2. N. Bourbaki, Elements of Mathematics. Commutative Algebra, Addison-Wesley Publishing Co., Reading, Mass., 1972.
  3. S. T. Dougherty and Y. H. Park, On modular cyclic codes, Finite Fields Appl. 13 (2007), no. 1, 31-57. https://doi.org/10.1016/j.ffa.2005.06.004
  4. P. Kanwar and S. R. Lopez-Permouth, Cyclic codes over the integers modulo $p^m$, Finite Fields Appl. 3 (1997), no. 4, 334-352. https://doi.org/10.1006/ffta.1997.0189
  5. R. Lidl and H. Niederreiter, Introduction to Finite Fields and Their Applications, Cambridge University Press, Cambridge, 1994.
  6. S. S. Woo, Cyclic codes of even length over $Z_4$, J. Korean Math. Soc. 44 (2007), no. 3, 697-706. https://doi.org/10.4134/JKMS.2007.44.3.697
  7. S. S. Woo, Free cyclic codes over finite local rings, Bull. Korean Math. Soc. 43 (2006), no. 4, 723-735. https://doi.org/10.4134/BKMS.2006.43.4.723
  8. S. S. Woo, Cyclic codes of length $2^n$ over $Z_4$, preprint, 2005.
  9. S. S. Woo, Algebras with a nilpotent generator over $Z_{p^2}$ , Bull. Korean Math. Soc. 43 (2006), no. 3, 487-497. https://doi.org/10.4134/BKMS.2006.43.3.487

Cited by

  1. CYCLIC CODES OF LENGTH 2nOVER ℤ4 vol.28, pp.1, 2013, https://doi.org/10.4134/CKMS.2013.28.1.039