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A generalization of Kung’s theorem

Abstract

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.

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References

  1. 1.

    Ball S., Blokhuis A.: A bound for the maximum weight of a linear code. SIAM J. Discret. Math. 27(1), 575–583 (2013).

  2. 2.

    Britz T., Shiromoto K.: On the covering dimension of a linear code. arXiv:1504.02357 (2015).

  3. 3.

    Crapo H., Rota G.-C.: On the Foundation of Combinatorial Theory: Combinatorial Geometries (Preliminary Edition). MIT Press, Cambridge (1970).

  4. 4.

    Johnsen T., Roksvold J.N., Verdure H.: A Generalization of Weight Polynomials to Matroids. arXiv:1311.6291 (2013).

  5. 5.

    Jurrius R.: Weight enumeration of codes from finite spaces. Des. Codes Cryptogr. 63(3), 321–330 (2012).

  6. 6.

    Jurrius R.: Codes, arrangements, matroids, and their polynomial links. PhD thesis, Technische Universiteit Eindhoven. http://alexandria.tue.nl/extra2/734704 (2012).

  7. 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).

  8. 8.

    Kløve T.: Support weight distribution of linear codes. Discret. Math. 106(107), 311–316 (1992).

  9. 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).

  10. 10.

    Wei V.K.: Generalized Hamming weights for linear codes. IEEE Trans. Inf. Theory 37(5), 1412–1418 (1991).

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Acknowledgments

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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Correspondence to Hugues Verdure.

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Communicated by D. Panario.

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Johnsen, T., Shiromoto, K. & Verdure, H. A generalization of Kung’s theorem. Des. Codes Cryptogr. 81, 169–178 (2016). https://doi.org/10.1007/s10623-015-0139-6

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Keywords

  • Linear code
  • Kung’s bound
  • Generalized Singleton bound

Mathematics Subject Classification

  • 94B05
  • 05E40