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

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Abstract

We previously proposed the first nontrivial examples of a code having support t-designs for all weights obtained from the Assmus–Mattson theorem and having support \(t'\)-designs for some weights with some \(t'>t\). This suggests the possibility of generalizing the Assmus–Mattson theorem, which is very important in design and coding theory. In the present paper, we generalize this example as a strengthening of the Assmus–Mattson theorem along this direction. As a corollary, we provide a new characterization of the extended Golay code \({\mathcal {G}}_{24}\).

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References

  1. Alltop W.O.: Extending \(t\)-designs. J. Comb. Theory Ser. A 18, 177–186 (1975).

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. 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  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  6. Bannai E., Miezaki T.: Toy models for D. H. Lehmer’s conjecture II. In: Quadratic and Higher Degree Forms, pp. 1–27. Springer, New York (2013).

  7. Bannai E., Miezaki T., Nakasora H.: A note on the Assmus–Mattson theorem for some binary codes II, in preparation.

  8. Bannai E., Miezaki T., Yudin V.A.: An elementary approach to toy models for Lehmer’s conjecture. (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 75(6), 3–16 (2011); translation in Izv. Math. 75(6), 1093–1106 (2011).

  9. Betsumiya K., Munemasa A.: On triply even binary codes. J. Lond. Math. Soc. 86(1), 1–16 (2012).

    Article  MathSciNet  Google Scholar 

  10. Cameron P.J., van Lint J.H.: Designs, Graphs, Codes and Their Links, London Mathematical Society Student Texts, vol. 22. Cambridge University Press, Cambridge (1991).

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  12. Höhn G.: Conformal designs based on vertex operator algebras. Adv. Math. 217–5, 2301–2335 (2008).

    Article  MathSciNet  Google Scholar 

  13. 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  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  15. 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 

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

    Article  MathSciNet  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  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  19. 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  Google Scholar 

  20. 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  Google Scholar 

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

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

  23. Wolfram Research, Inc., Mathematica, Version 11.2, Champaign (2017).

Download references

Acknowledgements

The authors thank Akihiro Munemasa for helpful discussions and computations in this research. Hiroyuki Nakasora thanks Masaaki Harada and Hiroki Shimakura for their comments in a seminar at Tohoku university. Tsuyoshi Miezaki is supported by JSPS KAKENHI (18K03217).

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Authors and Affiliations

  1. Faculty of Science and Engineering, Waseda University, Tokyo, 169-8555, Japan

    Tsuyoshi Miezaki

  2. Institute for Promotion of Higher Education, Kobe Gakuin University, Kobe, 651-2180, Japan

    Hiroyuki Nakasora

Authors
  1. Tsuyoshi Miezaki
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  2. Hiroyuki Nakasora
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Correspondence to Hiroyuki Nakasora.

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Communicated by V. D. Tonchev.

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Appendices

Appendix A Values of \(X_{ij}\)

$$\begin{aligned} X_{11}&=-2 d (d-n) \left( (n-2 d)^2-n+2\right) .\\ X_{12}&=-\frac{1}{4} (n-2) n^2.\\ X_{21}&=8 d^2 \left( {\begin{array}{c}n\\ 6\end{array}}\right) +\frac{1}{12} (n-1) n\\&\qquad \times \Big (-24 \left( {\begin{array}{c}n-2 d\\ 6\end{array}}\right) +16 d^5-32 d^4 n+24 d^4+24 d^3 n^2\\&\qquad -48 d^3 n+60 d^3-8 d^2 n^3+30 d^2 n^2-62 d^2 n+12 d^2\\&\qquad +d n^4-6 d n^3+17 d n^2-12 d n+8 d\Big )-8 d n \left( {\begin{array}{c}n\\ 6\end{array}}\right) +2 n^2 \left( {\begin{array}{c}n\\ 6\end{array}}\right) -2 n \left( {\begin{array}{c}n\\ 6\end{array}}\right) .\\ X_{22}&=\frac{1}{48} (n-1) n \left( n^3-6 n^2+8 n\right) -n \left( {\begin{array}{c}n\\ 6\end{array}}\right) .\\ X_{31}&=\frac{1}{12} \Big (-16 d^6 \left( {\begin{array}{c}n\\ 6\end{array}}\right) +32 d^5 n \left( {\begin{array}{c}n\\ 6\end{array}}\right) -16 d^5 \left( {\begin{array}{c}n\\ 6\end{array}}\right) -32 d^5 \left( {\begin{array}{c}n\\ 8\end{array}}\right) -24 d^4 n^2 \left( {\begin{array}{c}n\\ 6\end{array}}\right) \\&\quad +24 d^4 n \left( {\begin{array}{c}n\\ 6\end{array}}\right) -38 d^4 \left( {\begin{array}{c}n\\ 6\end{array}}\right) +64 d^4 n \left( {\begin{array}{c}n\\ 8\end{array}}\right) -48 d^4 \left( {\begin{array}{c}n\\ 8\end{array}}\right) +8 d^3 n^3 \left( {\begin{array}{c}n\\ 6\end{array}}\right) -8 d^3 n^2 \left( {\begin{array}{c}n\\ 6\end{array}}\right) \\&\quad -48 d^3 n^2 \left( {\begin{array}{c}n\\ 8\end{array}}\right) +16 d^3 n \left( {\begin{array}{c}n\\ 6\end{array}}\right) +52 d^3 \left( {\begin{array}{c}n\\ 6\end{array}}\right) +96 d^3 n \left( {\begin{array}{c}n\\ 8\end{array}}\right) -120 d^3 \left( {\begin{array}{c}n\\ 8\end{array}}\right) -d^2 n^4 \left( {\begin{array}{c}n\\ 6\end{array}}\right) \\&\quad -2 d^2 n^3 \left( {\begin{array}{c}n\\ 6\end{array}}\right) +16 d^2 n^3 \left( {\begin{array}{c}n\\ 8\end{array}}\right) +13 d^2 n^2 \left( {\begin{array}{c}n\\ 6\end{array}}\right) -60 d^2 n^2 \left( {\begin{array}{c}n\\ 8\end{array}}\right) -58 d^2 n \left( {\begin{array}{c}n\\ 6\end{array}}\right) +6 d^2 \left( {\begin{array}{c}n\\ 6\end{array}}\right) \\&\quad +124 d^2 n \left( {\begin{array}{c}n\\ 8\end{array}}\right) -24 d^2 \left( {\begin{array}{c}n\\ 8\end{array}}\right) +d n^4 \left( {\begin{array}{c}n\\ 6\end{array}}\right) -2 d n^4 \left( {\begin{array}{c}n\\ 8\end{array}}\right) -6 d n^3 \left( {\begin{array}{c}n\\ 6\end{array}}\right) +12 d n^3 \left( {\begin{array}{c}n\\ 8\end{array}}\right) \\&\quad +19 d n^2 \left( {\begin{array}{c}n\\ 6\end{array}}\right) -34 d n^2 \left( {\begin{array}{c}n\\ 8\end{array}}\right) -14 d n \left( {\begin{array}{c}n\\ 6\end{array}}\right) +12 d \left( {\begin{array}{c}n\\ 6\end{array}}\right) +24 d n \left( {\begin{array}{c}n\\ 8\end{array}}\right) -16 d \left( {\begin{array}{c}n\\ 8\end{array}}\right) \\&\quad +48 d \left( {\begin{array}{c}n\\ 6\end{array}}\right) \left( {\begin{array}{c}n-2 d\\ 6\end{array}}\right) +48 \left( {\begin{array}{c}n\\ 8\end{array}}\right) \left( {\begin{array}{c}n-2 d\\ 6\end{array}}\right) -48 \left( {\begin{array}{c}n\\ 6\end{array}}\right) \left( {\begin{array}{c}n-2 d\\ 8\end{array}}\right) \Big ).\\ X_{32}&=\frac{1}{192} \Big (n^4 (-\left( {\begin{array}{c}n\\ 6\end{array}}\right) )+12 n^3 \left( {\begin{array}{c}n\\ 6\end{array}}\right) -8 n^3 \left( {\begin{array}{c}n\\ 8\end{array}}\right) -44 n^2 \left( {\begin{array}{c}n\\ 6\end{array}}\right) +48 n^2 \left( {\begin{array}{c}n\\ 8\end{array}}\right) \\&\quad +48 n \left( {\begin{array}{c}n\\ 6\end{array}}\right) -64 n \left( {\begin{array}{c}n\\ 8\end{array}}\right) \Big ). \end{aligned}$$

Appendix B

See Table 1.

Table 1 Table of \((d^\perp ,t)=(4,1)\)
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Miezaki, T., Nakasora, H. A note on the Assmus–Mattson theorem for some binary codes. Des. Codes Cryptogr. 90, 1485–1502 (2022). https://doi.org/10.1007/s10623-022-01050-2

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