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One-weight and two-weight \(\mathbb {Z}_{2}\mathbb {Z}_{2}[u,v]\)-additive codes

  • Minjia ShiEmail author
  • Chenchen Wang
  • Rongsheng Wu
  • Yu Hu
  • Yaoqiang Chang
Article
  • 2 Downloads
Part of the following topical collections:
  1. Special Issue on Sequences and Their Applications

Abstract

In this paper, a class of additive codes which is referred to as \(\mathbb {Z}_{2}\mathbb {Z}_{2}[u,v]\)-additive codes is introduced. This is a generalization towards another direction of recently introduced \(\mathbb {Z}_{2}\mathbb {Z}_{4}\) codes (Doughterty et al., Appl. Algebra Eng. Commun. Comput. 27(2), 123–138, 2016). A MacWilliams-type identity that relates the weight enumerator of a code with its dual is proved. Furthermore, the structure and possible weights for all one-weight and two-weight \(\mathbb {Z}_{2}\mathbb {Z}_{2}[u,v]\)-additive codes are described. Additionally, we also construct some one-weight and two-weight \(\mathbb {Z}_{2}\mathbb {Z}_{2}[u,v]\)-additive codes to illustrate our obtained results.

Keywords

Additive codes One-weight codes Two-weight codes MacWilliams identity 

Mathematics Subject Classification (2010)

94B05 94B15 

Notes

References

  1. 1.
    Bahattin, Y., Karadeniz, S.: Linear codes over \(\mathbb {F}_{2}+ u\mathbb {F}_{2}+ v\mathbb {F}_{2}+ uv\mathbb {F}_{2}\). Des. Codes Crypt. 54(1), 61–81 (2010)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bonisoli, A.: Every equidistant linear code is a sequence of dual Hamming codes. Ars Combin. 18, 181–186 (1984)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Byrne, E., Greferath, M., Honold, T.: Ring geometries, two-weight codes, and strongly regular graphs. Des. Codes Crypt. 48(1), 1–16 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Calderbank, R., Kantor, W. M.: The geometry of two-weight codes. Bull. Lond. Math. Soc. 18(2), 97–122 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Carlet, C.: One-weight \(\mathbb {Z}_{4}\)-linear codes. Coding theory, cryptography and related areas, pp 57–72. Springer, Berlin (2000)CrossRefGoogle Scholar
  6. 6.
    Delsarte, P.: An algebraic approach to the association schemes of coding theory. Philips Res. Rep. Suppl. 10, 1–97 (1973)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Doughterty, S. T., Liu, H. W., Yu, L.: One weight \(\mathbb {Z}_{2}\mathbb {Z}_{4}\) additive codes. Appl. Algebra Eng. Commun. Comput. 27(2), 123–138 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Özen, M., Shi, M. J., Siap, V.: An identity between the m-spotty Rosenbloom-Tsfasman weight enumerators over finite commutative Frobenius rings. Bull. Korean Math. Soc. 52(3), 809–823 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Sari, M., Siap, V., Siap, I.: One-homogeneous weight codes over Finite chain rings. Bull. Korean Math 52(6), 2011–2023 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Shi, M. J.: Optimal p-ary codes from one-weight linear codes over \(\mathbb {Z}_{p^{m}}\). Chin. J. Electron. 22, 799–802 (2013)Google Scholar
  11. 11.
    Shi, M. J., Chen, L.: Construction of two-Lee weight codes over \(\mathbb {F}_{p}+v\mathbb {F}_{p}+v^{2}\mathbb {F}_{p}\). Int. J. Comput. Math. 93(3), 415–424 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Shi, M. J., Solé, P.: Optimal p-ary codes from one-weight codes and two-weight codes over \(\mathbb {F}_{p}+v\mathbb {F}_{p}\). J. Syst. Sci. Complex. 28(3), 679–690 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Shi, M. J., Solé, P., Wu, B.: Cyclic codes and the weight enumerator of linear codes over \(\mathbb {F}_{2}+v\mathbb {F}_{2}+v^{2}\mathbb {F}_{2}\). Appl. Comput. Math. 12(2), 247–255 (2013)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Shi, M. J., Wang, Y.: Optimal binary codes from one-Lee weight and two-Lee weight projective codes over \(\mathbb {Z}_{4}\). J. Syst. Sci. Complex. 27, 795–810 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Shi, M. J., Xu, L. L., Yang, G.: A note on one weight and two weight projective \(\mathbb {Z}_{4}\)-codes. IEEE Trans. Inf. Theory 63(1), 177–182 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Shi, M. J., Zhu, S. X., Yang, S. L.: A class of optimal p-ary codes from one-weight codes over \(\mathbb {F}_{p}[u]/\langle u^{m}\rangle \). J. Franklin Inst. 350, 929–937 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Wood, J. A.: The structure of linear codes of constant weight. Electron Notes Discrete Math. 354(3), 1007–1026 (2002)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Lu, Z, Zhu, S, Wang, L, et al.: One-Lee weight and two-Lee weight \(\mathbb {Z}_{2}\mathbb {Z}_{2}[u]\)-additive codes. arXiv:1609.09588v3 (2018)

Copyright information

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

Authors and Affiliations

  • Minjia Shi
    • 1
    Email author
  • Chenchen Wang
    • 1
  • Rongsheng Wu
    • 1
  • Yu Hu
    • 1
  • Yaoqiang Chang
    • 1
  1. 1.Key Laboratory of Intelligent Computing and Signal Processing of Ministry of Education, School of MathematicsAnhui UniversityAnhuiPeople’s Republic of China

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