Cryptography and Communications

, Volume 9, Issue 5, pp 637–646

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


DOI: 10.1007/s12095-016-0206-5

Cite this article as:
Shi, M., Wu, R., Liu, Y. et al. Cryptogr. Commun. (2017) 9: 637. doi:10.1007/s12095-016-0206-5


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 

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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