Designs, Codes and Cryptography

, Volume 81, Issue 1, pp 169–178 | Cite as

A generalization of Kung’s theorem

  • Trygve Johnsen
  • Keisuke Shiromoto
  • Hugues Verdure


We give a generalization of Kung’s theorem on critical exponents of linear codes over a finite field, in terms of sums of extended weight polynomials of linear codes. For all \(i=k+1,\ldots ,n\), we give an upper bound on the smallest integer m such that there exist m codewords whose union of supports has cardinality at least i.


Linear code Kung’s bound Generalized Singleton bound 

Mathematics Subject Classification

94B05 05E40 



The first and third author are grateful to Thomas Britz and the second author for sharing early versions of the joint manuscript [2] with all of us. The material and results there gave the inspiration for writing the present article. The second author was supported by JSPS KAKENHI Grant Number 14474695. The first author wants to thank IMPA, Rio de Janeiro, where he stayed while this work was completed. The authors would like to thank the reviewers for their valuable comments.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Trygve Johnsen
    • 1
  • Keisuke Shiromoto
    • 2
  • Hugues Verdure
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of TromsøTromsøNorway
  2. 2.Department of Mathematics and EngineeringKumamoto UniversityKumamotoJapan

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