Designs, Codes and Cryptography

, Volume 87, Issue 12, pp 3063–3075 | Cite as

Linear codes with small hulls in semi-primitive case

  • Claude Carlet
  • Chengju LiEmail author
  • Sihem Mesnager


The hull of a linear code is defined to be the intersection of the code and its dual, and was originally introduced to classify finite projective planes. The hull plays an important role in determining the complexity of algorithms for checking permutation equivalence of two linear codes and computing the automorphism group of a linear code. It has been shown that these algorithms are very effective in general if the size of the hull is small. It is clear that the linear codes with the smallest hull are LCD codes and with the second smallest hull are those with one-dimensional hull. In this paper, we employ character sums in semi-primitive case to construct LCD codes and linear codes with one-dimensional hull from cyclotomic fields and multiplicative subgroups of finite fields. Some sufficient and necessary conditions for these codes are obtained, where prime ideal decompositions of prime p in cyclotomic fields play a key role. In addition, we show the non-existence of these codes in some cases.


Linear code Hull LCD code Cyclotomic field Character sum 

Mathematics Subject Classification

94B05 11T24 11T71 



The authors are very grateful to the editor and the reviewers for their detailed comments and suggestions that much improved the presentation and quality of this paper. The work was supported by the National Natural Science Foundation of China under Grant 11701179, the Shanghai Chenguang Program under Grant 18CG22, the Shanghai Sailing Program under Grant 17YF1404300, the Foundation of State Key Laboratory of Integrated Services Networks under Grant ISN20-02, and the SECODE project in the scope of the CHIST-ERA Program.


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Copyright information

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

Authors and Affiliations

  • Claude Carlet
    • 1
    • 2
    • 3
  • Chengju Li
    • 4
    • 5
    Email author
  • Sihem Mesnager
    • 1
    • 2
    • 6
  1. 1.Department of MathematicsUniversity of Paris VIIISaint-DenisFrance
  2. 2.University of Paris XIII, CNRS, LAGA UMR 7539, Sorbonne Paris CitéVilletaneuseFrance
  3. 3.University of BergenBergenNorway
  4. 4.Shanghai Key Laboratory of Trustworthy ComputingEast China Normal UniversityShanghaiChina
  5. 5.State Key Laboratory of Integrated Services NetworksXidian UniversityXi’anChina
  6. 6.Telecom ParisTechParisFrance

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