Designs, Codes and Cryptography

, Volume 87, Issue 12, pp 3077–3086 | Cite as

Linear complementary dual codes over rings

  • Zihui LiuEmail author
  • Jinliang Wang


By using linear algebra over finite commutative rings, we will present some judging criterions for linear complementary dual (LCD) codes over rings, in particular, free LCD codes over finite commutative rings are described. By using free LCD codes over finite commutative rings and the Chinese Remainder Theorem, LCD codes over semi-simple rings are constructed and the equivalence of free codes and free LCD codes is given. In addition, all the possible LCD codes over chain rings are determined. We also generalize the judging criterion for cyclic LCD codes over finite fields to cyclic LCD codes over chain rings. Based on the above results and the Chinese Remainder Theorem, we also present results for LCD codes over principal ideal rings.


Principal ideal rings Chain rings LCD codes Chinese Remainder Theorem Generating polynomial 

Mathematics Subject Classification




  1. 1.
    Cao Y., Cao Y.L., Fu F.W.: Matrix-product structure of constacyclic codes over finite chain rings \({\mathbb{F}}_{p^m}[u]/\langle u^e\rangle \). Appl. Algebra Eng. Commun. Comput. 29(6), 455–478 (2018).CrossRefGoogle Scholar
  2. 2.
    Carlet C., Guilley S.: Complementary dual codes for counter-measures to side-channel attacks. Adv. Math. Commun. 10(1), 131–150 (2016).MathSciNetCrossRefGoogle Scholar
  3. 3.
    Carlet C., Mesnager S., Tang C.M., Qi Y.F., Pellikaam R.: Linear codes over \({\mathbb{F}}_q\) are equivalent to LCD codes for \(q>3\). IEEE Trans. Inf. Theory 64(4), 3010–3017 (2018).CrossRefGoogle Scholar
  4. 4.
    Dinh H.Q., Nguyen B.T., Sriboonchitta S.: Constacyclic codes over finite commutative semi-simple rings. Finite Fields Appl. 45, 1–18 (2017).MathSciNetCrossRefGoogle Scholar
  5. 5.
    Dougherty S.T., Kim J.L., Kulosman H.: MDS codes over finite principal ideal rings. Des. Codes Cryptogr. 50, 77–92 (2009).MathSciNetCrossRefGoogle Scholar
  6. 6.
    Jin L.F.: Construction of MDS codes with complementary duals. IEEE Trans. Inf. Theory 63(5), 2843–2847 (2017).MathSciNetzbMATHGoogle Scholar
  7. 7.
    Li C., Ding C., Li S.: LCD cyclic codes over finite fields. IEEE Trans. Inf. Theory 63(7), 4344–4356 (2017).MathSciNetCrossRefGoogle Scholar
  8. 8.
    Li S., Li C., Ding C., Liu H.: Two families of LCD BCH codes. IEEE Trans. Inf. Theory 63(9), 5699–5717 (2017).MathSciNetzbMATHGoogle Scholar
  9. 9.
    Liu X.S., Liu H.L.: LCD codes over finite chain rings. Finite Fields Appl. 34, 1–19 (2015).MathSciNetCrossRefGoogle Scholar
  10. 10.
    McDonald B.R.: Finite Rings with Identity. Marcel Dekker, New York (1974).zbMATHGoogle Scholar
  11. 11.
    McDonald B.R.: Linear Algebta Over Commutative Rings. Marcel Dekker Inc., New York (1984).Google Scholar
  12. 12.
    Mesnager S., Tang C., Qi Y.: Complementary dual algebraic geometry codes. IEEE Trans. Inf. Theory 64(4), 2390–2397 (2018).MathSciNetCrossRefGoogle Scholar
  13. 13.
    Norton G.H., Salagean A.: On the structure of linear and cyclic codes over a finite chain ring. Appl. Algebra Eng. Commun. Comput. 10, 489–506 (2000).MathSciNetCrossRefGoogle Scholar
  14. 14.
    Shi M.J., Zhang Y.P.: Quasi-twisted codes with constacyclic constituent codes. Finite Fields Appl. 39, 159–178 (2016).MathSciNetCrossRefGoogle Scholar
  15. 15.
    Shi M.J., Qian L.Q., Solé P.: On self-dual negacirculant codes of index two and four. Des. Codes Cryptogr. 86(11), 2485–2494 (2018).MathSciNetCrossRefGoogle Scholar
  16. 16.
    Shi M.J., Sok L., Solé P., Calkavur S.: Self-dual codes and orthogonal matrices over large finite fields. Finite Fields Appl. 54, 297–314 (2018).MathSciNetCrossRefGoogle Scholar
  17. 17.
    Sok L., Shi M.J., Solé P.: Construction of optimal LCD codes over large finite fields. Finite Fields Appl. 50, 138–153 (2018).MathSciNetCrossRefGoogle Scholar
  18. 18.
    Yang X., Massey J.L.: The condition for a cyclic code to have a complementary dual. Discret. Math. 126, 391–393 (1994).MathSciNetCrossRefGoogle Scholar

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

Authors and Affiliations

  1. 1.Department of MathematicsBeijing Institute of TechnologyBeijingChina
  2. 2.School of Computer ScienceLiaocheng UniversityLiaochengChina

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