Cryptography and Communications

, Volume 9, Issue 5, pp 637–646 | Cite as

Two and three weight codes over \(\mathbb {F}_{p}+u\mathbb {F}_{p}\)



We construct an infinite family of three-Lee-weight codes of dimension 2m, where m is singly-even, over the ring \(\mathbb {F}_{p}+u\mathbb {F}_{p}\) with u2=0. These codes are defined as trace codes. They have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using Gauss sums. By Gray mapping, we obtain an infinite family of abelian p-ary three-weight codes. When m is odd, and p≡3 (mod 4), we obtain an infinite family of two-weight codes which meets the Griesmer bound with equality. An application to secret sharing schemes is given.


Three-weight codes Two-weight codes Gauss sums Trace codes Secret sharing 

Mathematics Subject Classification (2010)

94B05 94B15 


  1. 1.
    Ashikmin, A., Barg, A.: Minimal vectors in linear codes. IEEE Trans. Inf. Theory 44(5), 2010–2017 (1998)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bonnecaze, A., Udaya, P.: Cyclic codes and self-dual codes over \(\mathbb {F}_{2}+u\mathbb {F}_{2}\). IEEE Trans. Inf. Theory 45(4), 1250–1255 (1999)CrossRefMATHGoogle Scholar
  3. 3.
    Calderbank, A.R., Goethals, J.M.: Three weight codes and association schemes. Philips J. Res. 39(4), 143–152 (1984)MathSciNetMATHGoogle Scholar
  4. 4.
    Courteau, B., Wolfmann, J.: On triple sum sets and three weight codes. Discret. Math. 50(2-3), 179–191 (1984)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Delsarte, P.: Weights of linear codes and strongly regular normed spaces. Discret. Math. 3(1-3), 47–64 (1972)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Ding, C., Li, C., Li, N., Zhou, Z.: Three weight cyclic codes and their weight distribution. Discret. Math. 339(2), 415–427 (2016)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Ding, C., Yang, J.: Hamming weights in irreducible cyclic codes. Discret. Math. 313(4), 434–446 (2013)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Ding, C., Yuan, J.: Covering and secret sharing with linear codes. Lect. Notes Comput. Sci. 2731, 11–25 (2003)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Ling, S., Solé, P.: Duadic codes over \(\mathbb {F}_{2}+u\mathbb {F}_{2}\). Appl. Algebra Eng. Commun. Comput. 12(5), 365–379 (2001)CrossRefGoogle Scholar
  10. 10.
    McEliece, R.J., Rumsey, H. Jr.: Euler products, cyclotomy, and coding. J. Number Theory 4(3), 302–311 (1972)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    MacWilliams, F.J., Sloane, N.J.A.: The theory of error-correcting codes, North-Holland (1977)Google Scholar
  12. 12.
    Wu, B., Zhu, S.X.: Trace codes over Galois extensions of ring \(\mathbb {F}_{2}+u\mathbb {F}_{2}\). J. Electron. Inf. Technol. 29, 2899–2901 (2007)Google Scholar
  13. 13.
    Yuan, J., Ding, C.: Secret sharing schemes from three classes of linear codes. IEEE Trans. Inf. Theory 52(1), 206–212 (2006)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Minjia Shi
    • 1
    • 2
    • 3
  • Rongsheng Wu
    • 3
  • Yan Liu
    • 3
  • Patrick Solé
    • 4
  1. 1.Key Laboratory of Intelligent Computing, Signal Processing, Ministry of EducationAnhui University No. 3 Feixi RoadHefeiPeople’s Republic of China
  2. 2.National Mobile Communications Research LaboratorySoutheast UniversityNanjingPeople’s Republic of China
  3. 3.School of Mathematical SciencesAnhui UniversityHefeiPeople’s Republic of China
  4. 4.CNRS/ LTCIUniversity Paris-SaclayParisFrance

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