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Designs, Codes and Cryptography

, Volume 87, Issue 1, pp 87–95 | Cite as

How many weights can a linear code have?

  • Minjia ShiEmail author
  • Hongwei Zhu
  • Patrick Solé
  • Gérard D. Cohen
Article

Abstract

We study the combinatorial function L(kq),  the maximum number of nonzero weights a linear code of dimension k over \({\mathbb {F}}_q\) can have. We determine it completely for \(q=2,\) and for \(k=2,\) and provide upper and lower bounds in the general case when both k and q are \(\ge 3.\) A refinement L(nkq),  as well as nonlinear analogues N(Mq) and N(nMq),  are also introduced and studied.

Keywords

Linear codes Hamming weight Perfect difference sets 

Mathematics Subject Classification

94B05 05B10 

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Key Laboratory of Intelligent Computing & Signal Processing, Ministry of EducationAnhui UniversityHefeiPeople’s Republic of China
  2. 2.School of Mathematical SciencesAnhui UniversityHefeiPeople’s Republic of China
  3. 3.CNRS/LAGA, University of Paris 8Saint-DenisFrance
  4. 4.TelecomParisTechParisFrance

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