Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

The primitive idempotents of irreducible constacyclic codes and LCD cyclic codes


In this paper, we investigate all irreducible factors of \(x^{l_{1}^{m_{1}}l_{2}^{m_{2}}} - a\) over \(\mathbb {F}_{q}\) and obtain all primitive idempotents in \(\mathbb {F}_{q}[x]/\langle x^{l_{1}^{m_{1}}l_{2}^{m_{2}}} - a \rangle \), where \(a \in \mathbb {F}_{q}^{*}\), l1, l2 are two distinct odd prime divisors of qt − 1 with \(\gcd (l_{1}l_{2},q(q-1))= 1\) for prime t. Furthermore, the weight distributions of all irreducible constacyclic codes of length \(l_{1}^{m_{1}}l_{2}^{m_{2}}\) are presented for t = 2. As an application, we determine all linear complementary dual cyclic codes of length \(l_{1}^{m_{1}}l_{2}^{m_{2}}\) over \(\mathbb {F}_{q}\).

This is a preview of subscription content, log in to check access.


  1. 1.

    Arora, S.K., Batra, S., Cohen, S.D., Pruthi, M.: The primitive idempotents of a cyclic group algebra. Southeast Asian Bull. Math. 26, 197–208 (2002)

  2. 2.

    Arora, S.K., Pruthi, M.: Minimal cyclic codes of length 2pn. Finite Fields Appl. 5, 177–187 (1999)

  3. 3.

    Bakshi, G.K., Raka, M.: Minimal cyclic codes of length pnq. Finite Fields Appl. 9, 432–448 (2003)

  4. 4.

    Batra, S., Arora, S.K.: Some cyclic codes of length 2pn. Des. Codes Cryptogr. 61, 41–69 (2011)

  5. 5.

    Carlet, C., Guilley, S.: Complementary dual codes for countermeasures to side-channel attacks. In: Coding Theory and Applications (CIM Series in Mathematical Sciences), vol. 3, pp 97–105. Springer (2014)

  6. 6.

    Carlet, C., Mesnager, S., Tang, C., Qi, Y., Pellikaan, R.: Linear codes over \(\mathbb {F}_{q}\) are equivalent to LCD codes for q > 3. IEEE Trans. Inf. Theory 64, 3010–3017 (2018)

  7. 7.

    Carlet, C., Mesnager, S., Tang, C., Qi, Y.: New characterization and parametrization of LCD codes. IEEE Trans. Inf. Theory. (2018).

  8. 8.

    Cassuto, Y., Blaum, M.: Codes for symbol-pair read channels. IEEE Trans. Inf. Theory 57, 8011–8020 (2011)

  9. 9.

    Chen, B., Liu, H., Zhang, G.: A class of minimal cyclic codes over finite fields. Des. Codes Cryptogr. 74, 285–300 (2015)

  10. 10.

    Chen, B., Lin, L., Liu, H.: Constacyclic symbol-pair codes: lower bounds and optimal constructions. IEEE Trans. Inf. Theory 63, 7661–7666 (2017)

  11. 11.

    Coulter, R.S.: Explicit evaluations of some Weil sums. Acta Arith. 83, 241–251 (1998)

  12. 12.

    Dinh, H.Q.: Structure of repeated-root cyclic and negacyclic codes of length 6ps and their duals. Contemp. Math. 609, 69–87 (2014)

  13. 13.

    Esmaeili, M., Yari, S.: On complementary-dual quasi-cyclic codes. Finite Fields Appl. 15, 375–386 (2009)

  14. 14.

    Ferraz, R., Polcino, C.M.: Idempotents in group algebras and minimal abelian codes. Finite Fields Appl. 13, 382–393 (2007)

  15. 15.

    Huffman, W.C., Pless, V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003)

  16. 16.

    Kai, X., Zhu, S., Li, P.: Constacyclic codes and some new quantum MDS codes. IEEE Trans. Inf. Theory 60, 2080–2086 (2014)

  17. 17.

    Kløve T.: Codes for Error Detection. World Scientific Publishing, Singapore (2007)

  18. 18.

    Li, F., Yue, Q.: The primitive idempotents and weight distributions of irreducible constacyclic codes. Des. Codes Cryptogr.

  19. 19.

    Li, F., Yue, Q., Li, C.: The minimum Hamming distances of irreducible cyclic codes. Finite Field Appl. 29, 225–242 (2014)

  20. 20.

    Li, C., Ding, C., Li, S.: LCD cyclic codes over finite fields. IEEE Trans. Inf. Theory 63, 4344–4356 (2017)

  21. 21.

    Lidl, R., Niederreiter, H.: Finite Fields. Cambridge University Press, Cambridge (2008)

  22. 22.

    Liu, X., Fan, Y., Liu, H.: Galois LCD codes over finite fields. Finite Fields Appl. 49, 227–242 (2018)

  23. 23.

    Massey, J. L.: Linear codes with complementary duals. Discrete Math. 337-342, 106 (1992)

  24. 24.

    Pruthi, M., Arora, S.K.: Minimal codes of prime-power length. Finite Fields Appl. 3, 99–113 (1997)

  25. 25.

    Sendrier, N.: Linear codes with complementary duals meet the Gilbert-Varshamov bound. Discrete Math. 285, 345–347 (2004)

  26. 26.

    Singh, R., Pruthi, M.: Primitive idempotents of irreducible quadratic residue cyclic codes of length pnqm. Int. J. Algebra 5, 285–294 (2011)

  27. 27.

    Wan, Z.: Lectures on Finite Fields and Galois Rings. World Scientific Publishing, Singapore (2003)

  28. 28.

    Wu, Y., Yue, Q.: Factorizations of binomial polynomials and enumerations of LCD and self-dual constacyclic codes. IEEE Trans. Inf. Theory. (2018).

  29. 29.

    Yang, X., Massey, J. L.: The necessary and sufficient condition for a cyclic code to have a complementary dual. Discrete Math. 126, 391–393 (1994)

  30. 30.

    Zhou, Z., Tang, C., Li, X., Ding, C: Binary LCD codes and self-orthogonal codes from a generic construction. IEEE Trans. Inf. Theory. (2018).

Download references


The authors are very grateful to the reviewers and the Editor for their comments and suggestions that improved the quality and presentation of this paper. This research is supported by the National Natural Science Foundation of China under Grant No. 61571243.

Author information

Correspondence to Zexia Shi.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Shi, Z., Fu, F. The primitive idempotents of irreducible constacyclic codes and LCD cyclic codes. Cryptogr. Commun. 12, 29–52 (2020).

Download citation


  • Primitive idempotent
  • Irreducible constacyclic code
  • Weight distribution
  • LCD code

Mathematics Subject Classification (2010)

  • 11T71
  • 94B05
  • 94B15