A generalization of Kung’s theorem
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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.
KeywordsLinear code Kung’s bound Generalized Singleton bound
Mathematics Subject Classification94B05 05E40
The first and third author are grateful to Thomas Britz and the second author for sharing early versions of the joint manuscript  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.
- 1.Ball S., Blokhuis A.: A bound for the maximum weight of a linear code. SIAM J. Discret. Math. 27(1), 575–583 (2013).Google Scholar
- 2.Britz T., Shiromoto K.: On the covering dimension of a linear code. arXiv:1504.02357 (2015).
- 3.Crapo H., Rota G.-C.: On the Foundation of Combinatorial Theory: Combinatorial Geometries (Preliminary Edition). MIT Press, Cambridge (1970).Google Scholar
- 4.Johnsen T., Roksvold J.N., Verdure H.: A Generalization of Weight Polynomials to Matroids. arXiv:1311.6291 (2013).
- 5.Jurrius R.: Weight enumeration of codes from finite spaces. Des. Codes Cryptogr. 63(3), 321–330 (2012).Google Scholar
- 6.Jurrius R.: Codes, arrangements, matroids, and their polynomial links. PhD thesis, Technische Universiteit Eindhoven. http://alexandria.tue.nl/extra2/734704 (2012).
- 7.Jurrius R., Pellikaan R.: Codes, arrangements and matroids. In: Algebraic Geometry Modeling in Information Theory. Series on Coding Theory and Cryptology, vol. 8, pp. 219–325. World Scientific Publishing, Hackensack (2013).Google Scholar
- 8.Kløve T.: Support weight distribution of linear codes. Discret. Math. 106(107), 311–316 (1992).Google Scholar
- 9.Kung, J.P.S.: Critical problems. In: Matroid Theory (Seattle, WA, 1995). Contemporary Mathematics, vol. 197, pp. 1–127. American Mathematical Society, Providence (1996).Google Scholar
- 10.Wei V.K.: Generalized Hamming weights for linear codes. IEEE Trans. Inf. Theory 37(5), 1412–1418 (1991).Google Scholar