On self-dual and LCD double circulant and double negacirculant codes over \(\mathbb {F}_{q}+u\mathbb {F}_{q}\)

  • Minjia ShiEmail author
  • Hongwei Zhu
  • Liqin Qian
  • Lin Sok
  • Patrick Solé


Double circulant codes of length 2n over the non-local ring \(R=\mathbb {F}_{q}+u\mathbb {F}_{q}, u^{2}=u,\) are studied when q is an odd prime power, and − 1 is a square in \(\mathbb {F}_{q}\). Double negacirculant codes of length 2n are studied over R when n is even, and q is an odd prime power. Exact enumeration of self-dual and LCD such codes for given length 2n is given. Employing a duality-preserving Gray map, self-dual and LCD codes of length 4n over \(\mathbb {F}_{q}\) are constructed. Using random coding and the Artin conjecture, the relative distance of these codes is bounded below for n. The parameters of examples of modest lengths are computed. Several such codes are optimal.


Double circulant codes Double negacirculant codes Codes over rings Self-dual codes LCD codes Artin conjecture 

Mathematics Subject Classification (2010)

94 B15 94 B25 05 E30 



This research is supported by National Natural Science Foundation of China (61672036) and Excellent Youth Foundation of Natural Science Foundation of Anhui Province (1808085J20).


  1. 1.
    Alahmadi, A., Güneri, C., Özkaya, B., Shoaib, H., Solé, P.: On self-dual double negacirculant codes. Discret. Appl. Math. 222, 205–212 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alahmadi, A., Güneri, C., Özkaya, B., Shoaib, H., Solé, P.: On linear complementary-dual multinegacirculant codes. arXiv:1703.03115v1 [cs.IT] (2017)
  3. 3.
    Alahmadi, A., Ozdemir, F., Solé, P.: On self-dual double circulant codes. Designs Codes & Cryptography. (2017)
  4. 4.
    Dougherty, S.T., Gaborit, P., Harada, M., Munemasa, A., Solé, P.: Type IV codes over rings. IEEE Trans. Inf. Theory 45(7), 2345–2360 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Dougherty, S.T., Kim, J.L., Özkaya, B., Sok, L., Solé, P.: The combinatorics of LCD codes: Linear programming bound and orthogonal matrices. Int. J. Inf. Coding Theory 4(2/3), 116–128 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Grassl, M.: Bounds on the minimum distance of linear codes and quantum codes. Online available at
  7. 7.
    Güneri, C., Özkaya, B., Solé, P.: Quasi-cyclic complementary dual codes. Finite Fields Their Appl. 42, 67–80 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hooley, C.: On Artin’s conjecture. J. Reine Angew. Math 225, 209–220 (1967)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Huffman, W.C., Pless, V: Fundamentals of Error Correcting Codes. Cambridge University Press (2003)Google Scholar
  10. 10.
    Jia, Y.: On quasi-twisted codes over finite fields. Finite Fields Appl. 18, 237–257 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lidl, R., Niederreiter, H.: Finite Fields. Addison-Wesley, Reading (1983)zbMATHGoogle Scholar
  12. 12.
    Ling, S., Solé, P.: On the algebraic structure of quasi-cyclic codes I: Finite fields. IEEE Trans. Inf. Theory 47(7), 2751–2760 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ling, S., Solé, P.: On the algebraic structure of quasi-cyclic codes II: Chain rings. Des. Codes Cryptogr. 30(1), 113–130 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Liu, Y., Shi, M.J., Solé, P.: Construction of hermitian self-dual constacyclic codes over \(\mathbb {F}_{q^{2}}+v\mathbb {F}_{q^{2}}\). Appl. Comput. Math. 15(3), 359–369 (2016)MathSciNetGoogle Scholar
  15. 15.
    Meyn, H.: Factorization of the cyclotomic polynomial \(x^{2^{n}}+ 1\) over finite fields. Finite Fields Appl. 2, 439–442 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Moree, P.: Artin’s primitive root conjecture a survey. Integers 10(6), 1305–1416 (2012)MathSciNetzbMATHGoogle Scholar
  17. 17.
  18. 18.
    Shi, M.J., Zhu, H.W., Solé, P.: On the self-dual four-circulant codes. Int. J. Found. Comput. Sci. 29(7), 1143–1150 (2018)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Shi, M.J., Guan, Y., Solé, P.: Two new families of two-weight codes. IEEE Trans. Inf. Theory 63(10), 6240–6246 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Zhu, S.X., Wang, L.: A class of constacyclic codes over \(\mathbb {F},_{p}+v\mathbb {F}_{p}\). Discret. Math. 311, 2677–2682 (2011)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Minjia Shi
    • 1
    Email author
  • Hongwei Zhu
    • 1
  • Liqin Qian
    • 1
  • Lin Sok
    • 1
  • Patrick Solé
    • 2
  1. 1.School of Mathematical SciencesAnhui UniversityHefeiChina
  2. 2.Aix Marseille Univ, CNRS, Centrale Marseille, I2MMarseilleFrance

Personalised recommendations