Skip to main content
SpringerLink
Log in

A note on the Assmus–Mattson theorem for some binary codes II

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

Let C be a four-weight binary code, which has all one vector. Furthermore, we assume that C supports t-designs for all weights obtained from the Assmus–Mattson theorem. We previously showed that \(t\le 5\). In the present paper, we show an analogue of this result in the cases of five and six-weight codes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alekseyev M.: When do binomial coefficients sum to a power of 2? Mathoverflow. https://mathoverflow.net/questions/412940/

  2. Assmus E.F. Jr., Mattson H.F. Jr.: New \(5\)-designs. J. Comb. Theory 6, 122–151 (1969).

    Article  MathSciNet  MATH  Google Scholar 

  3. Bachoc C.: On harmonic weight enumerators of binary codes. Des. Codes Cryptogr. 18(1–3), 11–28 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  4. Bannai E., Miezaki T.: Toy models for D. H. Lehmer’s conjecture. J. Math. Soc. Jpn. 62(3), 687–705 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  5. Bannai E., Miezaki T.: Toy models for D. H. Lehmer’s conjecture II: quadratic and higher degree forms. Dev. Math. 31, 1–27 (2013).

    MATH  Google Scholar 

  6. Bannai E., Koike M., Shinohara M., Tagami M.: Spherical designs attached to extremal lattices and the modulo p property of Fourier coefficients of extremal modular forms. Mosc. Math. J. 6, 225–264 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  7. Bannai E., Miezaki T., Yudin V.A.: An elementary approach to toy models for Lehmer’s conjecture. Russ. Izv. Ross. Akad. Nauk Ser. Mat. 75(6), 3–16 (2011).

    MathSciNet  MATH  Google Scholar 

  8. Bannai E., Miezaki T., Yudin V.A.: An elementary approach to toy models for Lehmer’s conjecture. Russ. Izv. Math. 75(6), 1093–1106 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  9. Bosma W., Cannon J., Playoust C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24, 235–265 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  10. Delsarte P.: Hahn polynomials, discrete harmonics, and \(t\)-designs. SIAM J. Appl. Math. 34(1), 157–166 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  11. Horiguchi N., Miezaki T., Nakasora H.: On the support designs of extremal binary doubly even self-dual codes. Des. Codes Cryptogr. 72, 529–537 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  12. Lehmer D.H.: The vanishing of Ramanujan’s \(\tau (n)\). Duke Math. J. 14, 429–433 (1947).

    Article  MathSciNet  MATH  Google Scholar 

  13. Macwilliams J.: A theorem on the distribution of weights in a systematic code. Bell Syst. Tech. J. 42, 79–84 (1963).

    Article  MathSciNet  Google Scholar 

  14. Miezaki T., Nakasora H.: On the Assmus–Mattson type theorem for Type I and even formally self-dual codes, submitted.

  15. Miezaki T.: Tsuyoshi Miezaki’s website. http://www.f.waseda.jp/miezaki/data.html.

  16. Miezaki T.: Conformal designs and D. H. Lehmer’s conjecture. J. Algebra 374, 59–65 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  17. Miezaki T.: Design-theoretic analogies between codes, lattices, and vertex operator algebras. Des. Codes Cryptogr. 89(5), 763–780 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  18. Miezaki T., Nakasora H.: An upper bound of the value of \(t\) of the support \(t\)-designs of extremal binary doubly even self-dual codes. Des. Codes Cryptogr. 79, 37–46 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  19. Miezaki T., Nakasora H.: The support designs of the triply even binary codes of length \(48\). J. Comb. Des. 27, 673–681 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  20. Miezaki T., Nakasora H.: A note on the Assmus–Mattson theorem for some binary codes. Des. Codes Cryptogr. 90(6), 1485–1502 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  21. Miezaki T., Munemasa A., Nakasora H.: A note on Assmus–Mattson type theorems. Des. Codes Cryptogr. 89, 843–858 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  22. Venkov B.B.: Even unimodular extremal lattices (Russian), Algebraic geometry and its applications (Translation in Proc. Steklov Inst. Math. 165 (1985) 47–52). Trudy Mat. Inst. Steklov. 165, 43–48 (1984).

    MathSciNet  Google Scholar 

  23. Venkov B.B.: Réseaux et designs sphériques, (French) [Lattices and spherical designs] Réseaux euclidiens, designs sphériques et formes modulaires. Monogr. Enseign. Math. 37, 10–86 (2001).

    MathSciNet  MATH  Google Scholar 

  24. Wolfram Research, Inc.: Mathematica, Version 11.2, Champaign, IL (2017).

Download references

Acknowledgements

The authors would like to thank the anonymous reviewers for their beneficial comments on an earlier version of the manuscript. The second named author was supported by JSPS KAKENHI (22K03277). We thank Maxine Garcia, PhD, from Edanz (https://jp.edanz.com/ac) for editing a draft of this manuscript.

Author information

Authors and Affiliations

  1. Faculty of Mathematics, Kyushu University (emeritus), Fukuoka, Japan

    Eiichi Bannai

  2. Faculty of Science and Engineering, Waseda University, Tokyo, 169–8555, Japan

    Tsuyoshi Miezaki

  3. Institute for Promotion of Higher Education, Kobe Gakuin University, Kobe, 651–2180, Japan

    Hiroyuki Nakasora

Authors
  1. Eiichi Bannai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tsuyoshi Miezaki
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hiroyuki Nakasora
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hiroyuki Nakasora.

Additional information

Communicated by V. D. Tonchev.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bannai, E., Miezaki, T. & Nakasora, H. A note on the Assmus–Mattson theorem for some binary codes II. Des. Codes Cryptogr. 91, 2509–2522 (2023). https://doi.org/10.1007/s10623-023-01212-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-023-01212-w

Keywords

Mathematics Subject Classification

Search

Navigation