Extremal invariant polynomials not satisfying the Riemann hypothesis

  • Koji ChinenEmail author
Original Paper


Zeta functions for linear codes were defined by Iwan Duursma in 1999 as generating functions for the Hamming weight enumerators of linear codes. They were generalized to the case of some invariant polynomials and so-called the formal weight enumerators by the present author. One of the most important problems from the applicable point of view is whether extremal weight enumerators satisfy the Riemann hypothesis. In this article, we show there exist extremal polynomials of the weight enumerator type, not being related to existing codes, which are invariant under the MacWilliams transform and do not satisfy the Riemann hypothesis. Such examples are contained in a certain invariant polynomial ring which is found by the use of the binomial moment. We determine the group which fixes the ring. To formulate the extremal property, we also establish an analog of the Mallows–Sloane bound for a certain sequence of the members in the ring.


Invariant polynomial ring Extremal weight enumerator Zeta function for codes Riemann hypothesis Binomial moment 

Mathematics Subject Classification

Primary 11T71 Secondary 13A50 12D10 



This work was supported by Japan Society for the Promotion of Science KAKENHI Grant No. JP26400028. This work was established mainly during the author’s stay at University of Strasbourg for the overseas research program of Kindai University in 2016. He would like to express his sincere gratitude to Professor Yann Bugeaud at University of Strasbourg for his hospitality and to Kindai University for giving him a chance of the program.


  1. 1.
    Chinen, K.: Zeta functions for formal weight enumerators and the extremal property. Proc. Jpn. Acad. Ser. A. 81, 168–173 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chinen, K.: An abundance of invariant polynomials satisfying the Riemann hypothesis. Discrete Math. 308, 6426–6440 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chinen, K.: Construction of divisible formal weight enumerators and extremal polynomials not satisfying the Riemann hypothesis. arXiv:1709.03380
  4. 4.
    Duursma, I.: Weight distribution of geometric Goppa codes. Trans. Am. Math. Soc. 351(9), 3609–3639 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Duursma, I.: From weight enumerators to zeta functions. Discrete Appl. Math. 111, 55–73 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Duursma, I.: A Riemann hypothesis analogue for self-dual codes. DIMACS Ser. Discrete Math. Theor. Comput. Sci. 56, 115–124 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Duursma, I.: Extremal weight enumerators and ultraspherical polynomials. Discrete Math. 268(1–3), 103–127 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. Elsevier, Amsterdam (1977)zbMATHGoogle Scholar
  9. 9.
    Pless, V.S.: Introduction to the Theory of Error-Correcting Codes, 3rd edn. Wiley, NewYork (1998)CrossRefzbMATHGoogle Scholar
  10. 10.
    Sloane, N.J.A.: Self-dual codes and lattices. In: Relations Between Combinatorics and Other Parts of Mathematics, Proceedings of Symposium on Pure Mathematics 34. Ohio State University, Columbus (1978), American Mathematical Society, Providence, pp. 273–208 (1979)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, School of Science and EngineeringKindai UniversityHigashi-OsakaJapan

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