DOI QR코드

DOI QR Code

FREE CYCLIC CODES OVER FINITE LOCAL RINGS

  • Published : 2006.11.30

Abstract

In [2] it was shown that a 1-generator quasi-cyclic code C of length n = ml of index l over $\mathbb{Z}_4$ is free if C is generated by a polynomial which divides $X^m-1$. In this paper, we prove that a necessary and sufficient condition for a cyclic code over $\mathbb{Z}_pk$ of length m to be free is that it is generated by a polynomial which divides $X^m-1$. We also show that this can be extended to finite local rings with a principal maximal ideal.

Keywords

free modules over a finite commutative rings;separable extension of local rings;cyclic codes over $\mathbb{Z}_pk$

References

  1. M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley, 1969
  2. N. Aydin and D. K. Ray-Chaudhuri, Quasi-cyclic codes over $Z_{4}$ and some new binary codes, IEEE Trans. Inform. Theory 48 (2002), no. 7, 2065-2069 https://doi.org/10.1109/TIT.2002.1013145
  3. N. Bourbaki, Elements of Mathematics, Commutative Algebra, Addison-Wesley, 1972
  4. P. Kanwar and S. R. Lopez-Permouth, Cyclic codes over the integers modulo $P^{n}$ , Finite Fields Appl. 3 (1997), no. 4, 334-352 https://doi.org/10.1006/ffta.1997.0189
  5. S. Ling and P. Sole, On the algebraic structure of cyclic code I: finite fields, IEEE Transactions on Information Theory 47 (2001), no. 7, 2751-2760 https://doi.org/10.1109/18.959257
  6. B. R. McDonald, Finite rings with identity, Marcel Dekker, 1974. Department of Mathematics, Ewha Women's University, Seoul 120-750

Cited by

  1. On quasi-cyclic codes over $${\mathbb{Z}_q}$$ vol.20, pp.5-6, 2009, https://doi.org/10.1007/s00200-009-0110-8
  2. IDEALS OF Zpn[X]/(Xl-1) vol.26, pp.3, 2011, https://doi.org/10.4134/CKMS.2011.26.3.427
  3. CYCLIC CODES OF LENGTH 2nOVER ℤ4 vol.28, pp.1, 2013, https://doi.org/10.4134/CKMS.2013.28.1.039