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


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.


Linear codes Hamming weight Perfect difference sets 

Mathematics Subject Classification

94B05 05B10 


  1. 1.
    Alberson T., Neri A.: Maximum weight spectrum codes.
  2. 2.
    Baker R.C., Harman G., Pintz J.: The difference between consecutive primes, II. Proc. Lond. Math. Soc. 83(3), 532–562 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Beth T., Jungnickel D., Lenz H.: Design theory. BI-Institut, Mannheim, Wien, Zurich (1985).Google Scholar
  4. 4.
    Cohen G., Honkala I., Litsyn S., Solé P.: Long packing and covering codes. IEEE Trans. Inf. Theory 43, 1617–1619 (1997).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Delsarte P.: Four fundamentals parameters of a code and their combinatorial significance. Inf. Control 23, 407–438 (1973).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Enomoto H., Frankl P., Ito N., Nomura K.: Codes with given distances. Graph Comb. 3, 25–38 (1987).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Huffman W.C., Pless V.: Fundamentals of Error Correcting Codes. Cambridge University Press, Cambridge (2003).CrossRefzbMATHGoogle Scholar
  8. 8.
  9. 9.
    Singer J.: A theorem in finite geometry and some applications to number theory. Trans. Am. Math Soc. 43, 377–385 (1938).MathSciNetCrossRefzbMATHGoogle Scholar

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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