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Mathematics in Computer Science

, Volume 6, Issue 2, pp 109–120 | Cite as

Relations Between Möbius and Coboundary Polynomials

  • Relinde Jurrius
Open Access
Article

Abstract

It is known that, in general, the coboundary polynomial and the Möbius polynomial of a matroid do not determine each other. Less is known about more specific cases. In this paper, we will investigate if it is possible that the Möbius polynomial of a matroid, together with the Möbius polynomial of the dual matroid, define the coboundary polynomial of the matroid. In some cases, the answer is affirmative, and we will give two constructions to determine the coboundary polynomial in these cases.

Keywords

Matroid theory Möbius polynomial Coboundary polynomial Coding theory 

Mathematics Subject Classification (2010)

05B35 11T71 

Notes

Acknowledgments

The author would like to thank Ruud Pellikaan for stating the Main Question, and for valuable comments on this paper. The author is also indebted to an anonymous referee for the various comments on the paper and for the material in Sect. 6.

Open Access

This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

References

  1. 1.
    Aigner M.: Combinatorial Theory. Springer, New York (1979)zbMATHCrossRefGoogle Scholar
  2. 2.
    Brylawski, T.H.: The Tutte polynomial. I. General theory. In C.I.M.E. Summer Schools (1980)Google Scholar
  3. 3.
    Crapo H.: The Tutte polynomial. Aequat. Math. 3, 211–229 (1969)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    de Boer M.A.: Almost MDS codes. Des. Codes Cryptogr. 9, 143–155 (1996)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Duursma I.M.: Weight distributions of geometric Goppa codes. Trans. Am. Math. Soc. 351, 3609–3639 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Duursma I.M.: From weight enumerators to zeta functions. Discrete Appl. Math. 111, 55–73 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Duursma, I.M.: Combinatorics of the two-variable zeta function. In: Mullen, G.L., Poli, A., Stichtenoth, H. (eds.) International Conference on Finite Fields and Applications. In: Lecture Notes in Computer Science, vol. 2948, pp. 109–136. Springer, Berlin (2003)Google Scholar
  8. 8.
    Duursma I.M.: Extremal weight enumerators and ultraspherical polynomials. Discrete Math. 268, 103–127 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Faldum A., Willems W.: Codes of small defect. Des. Codes Cryptogr. 10(3), 341–350 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Greene C.: Weight enumeration and the geometry of linear codes. Stud. Appl. Math. 55, 119–128 (1976)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Jurrius R.P.M.J., Pellikaan R.: Codes, arrangements and matroids. In: Series on Coding Theory and Cryptology. World Scientific Publishing, Singapore (2011)Google Scholar
  12. 12.
    Kløve T.: The weight distribution of linear codes over GF(q l) having generator matrix over GF(q). Discrete Math. 23, 159–168 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    MacWilliams F.J., Sloane N.J.A.: The theory of error-correcting codes. North-Holland Mathematical Library, Amsterdam (1977)zbMATHGoogle Scholar
  14. 14.
    Oxley J.G.: Matroid Theory, 2nd edn. Oxford University Press, Oxford (2011)zbMATHGoogle Scholar
  15. 15.
    Stanley, R.P.: An introduction to hyperplane arrangements. In: Geometric Combinatorics, IAS/Park City Math. Ser., vol. 13, pp. 389–496. American Mathematical Society, Providence, (2007)Google Scholar
  16. 16.
    Welsh D.J.A.: Matroid Theory. Academic Press, London (1976)zbMATHGoogle Scholar
  17. 17.
    White N.: Theory of matroids. In: Encyclopedia of Mathmatics and its Applications, vol. 26. Cambridge University Press, Cambridge (1986)Google Scholar
  18. 18.
    Zaslavsky, T.: Facing up to arrangements: face-count fomulas for partitions of space by hyperplanes. Memoirs of American Mathematical Society, vol. 1, No. 154. American Mathematical Society, Providence (1975)Google Scholar
  19. 19.
    Zaslavsky T.: Signed graph colouring. Discrete Math. 39, 215–228 (1982)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© The Author(s) 2012

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceEindhoven University of TechnologyEindhovenThe Netherlands

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