\((1+\lambda u^2)\)-constacyclic codes of arbitrary length over \(F_{p^m}[u]/\langle u^3\rangle \)

Original Research
First Online:
Received:
  • 23 Downloads

Abstract

The generators for \((1+\lambda u^2)\)-constacyclic codes of arbitrary length over \(F_{p^m}[u]/\langle u^3\rangle \) are given. As a special case, we get generators for \((1+\lambda u^2)\)-constacyclic codes with length \(p^e\) over \(F_{p^m}[u]/\langle u^3\rangle \). Besides, we determine generators for the dual codes of \((1+\lambda u^2)\)-constacyclic codes with an arbitrary length over \(F_{p^m}[u]/\langle u^3\rangle \). Based on this, the necessary and sufficient condition for being a self-dual \((1+\lambda u^2)\)-constacyclic code over \(F_{2^m}[u]/\langle u^3\rangle \) is derived.

Keywords

Constacyclic code Cyclic code Self-dual code Generator 
This is a preview of subscription content, log in to check access

References

  1. 1.
    Qian, J., Zhang, L., Zhu, S.: \((1+u)\)-Constacyclic and cyclic codes over \(F_2+uF_2\). Appl. Math. Lett. 19(8), 820–823 (2006)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Shi, M., Zhu, S.: Constacyclic codes over ring \(F_q+uF_q+\cdots +u^{k-1}F_q\). J. Univ. Sci. Technol. China 39(6), 584–587 (2009)Google Scholar
  3. 3.
    Kai, X., Zhu, S., Li, P.: \((1+\lambda u)\)-Constacyclic codes over \(F_p[u]/\langle u^k\rangle \). J. Frankl. Inst. 347, 751–762 (2010)CrossRefMATHGoogle Scholar
  4. 4.
    Ding, J., Li, H.: The gray images of \((1+u)\)-constacyclic codes over \(F_{2^m}[u]/\langle u^k\rangle \). J. Appl. Math. Comput. 49, 433–445 (2015)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Udomkavanich, P., Jitman, S.: On the gray image of \((1+u^m)\)-cyclic codes over \(F_{p^k}+uF_{p^k}+\cdots +u^{m}F_{p^k}\). Int. J. Contemp. Math. Sci. 4(26), 1265–1272 (2009)MathSciNetMATHGoogle Scholar
  6. 6.
    Sobhani, R.: Complete classification of \((\delta +\alpha u^2)\)-constacyclic codes of length \(p^k\) over \(F_{p^m}+uF_{p^m}+u^2F_{p^m}\). Finite Fields Appl. 34, 123–138 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Sobhani, R., Sun, Z., Wang, L., et al.: A note on complete classification of \((\delta +\alpha u^2)\)-constacyclic codes of length \(p^k\) over \(F_{p^m}+uF_{p^m}+u^2F_{p^m}\). arXiv:1702.04694V1. Accessed 15 Feb 2017
  8. 8.
    Al-Ashker, M.M., Chen, J.: Cyclic codes of arbitrary lengths over \(F_q+uF_q+\cdots +u^{k-1}F_q\). Palest. J. Math. 2(1), 72–80 (2013)MathSciNetGoogle Scholar
  9. 9.
    Han, M., Ye, Y.P., Zhu, S., et al.: Cyclic codes over \(R=F_{p^m}+uF_{p^m}+\cdots +u^{k-1}F_{p^m}\) with length \(p^sn\). Inf. Sci. 181(4), 926–934 (2011)CrossRefMATHGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2018

Authors and Affiliations

  1. 1.Department of General EducationAnhui Xinhua UniversityHefeiPeople’s Republic of China

Personalised recommendations