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Design-theoretic analogies between codes, lattices, and vertex operator algebras

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Abstract

There are many analogies between codes, lattices, and vertex operator algebras. For example, extremal objects are good examples of combinatorial, spherical, and conformal designs. In this study, we investigated these objects from the aspect of design theory.

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Notes

  1. Prof. Shimakura pointed us to the following: It can be proved that the homogeneous spaces of \(V_{\sqrt{2}{\mathbb {Z}}^{2}}\) are conformal 3-designs and not conformal 4-designs. Moreover, the homogeneous spaces of \(V_{A_{2}}\) are not conformal 6-designs [29].

References

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

    Article  MathSciNet  Google Scholar 

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

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

    Article  MathSciNet  Google Scholar 

  5. Bannai E., Miezaki T.: Toy models for D. H. Lehmer’s conjecture II. In: Quadratic and Higher Degree Forms, pp. 1–27. Dev. Math., vol. 31. Springer, New York (2013).

  6. Borcherds R.E.: Vertex algebras, Kac-Moody algebras, and the Monster. Proc. Natl. Acad. Sci. USA 83, 3068–3071 (1986).

    Article  MathSciNet  Google Scholar 

  7. Conway J.H., Sloane N.J.A.: Sphere Packing, Lattices and Groups, 3rd edn. Springer-Verlag, New York (1999).

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. Delsarte P., Goethals J.-M., Seidel J.J.: Spherical codes and designs. Geom. Dedicata 6, 363–388 (1977).

    Article  MathSciNet  Google Scholar 

  10. Dong C., Griess R.L.: Rank one lattice type vertex operator algebras and their automorphism groups. J. Algebra 208, 262–275 (1998).

    Article  MathSciNet  Google Scholar 

  11. Dong C., Li H., Mason G.: Twisted representations of vertex operator algebras. Math. Ann. 310(3), 571–600 (1988).

    Article  MathSciNet  Google Scholar 

  12. Dong C., Mason G., Nagatomo K.: Quasi-modular forms and trace functions associated to free boson and lattice vertex operator algebras. Internat. Math. Res. Notices 8, 409–427 (2001).

    Article  MathSciNet  Google Scholar 

  13. Frenkel I., Huang Y., Lepowsky J.: On axiomatic approaches to vertex operator algebras and modules. Mem. Am. Math. Soc. 104, 4941 (1993).

    MathSciNet  MATH  Google Scholar 

  14. Frenkel I.B., Lepowsky J., Meurman A.: Vertex Operator Algebras and the Monster, vol. 134. Pure Appl. Math. Academic Press, Boston (1988).

    MATH  Google Scholar 

  15. Han G.-N., Ono K.: Hook lengths and 3-cores. Ann. Comb. 15, 305–312 (2011).

    Article  MathSciNet  Google Scholar 

  16. Höhn G.: Selbstduale Vertexoperatorsuperalgebren und das Babymonster. PhD thesis, Universität Bonn, 1995 Bonner Math. Schriften, vol. 286 (1996).

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  19. Mallows C.L., Odlyzko A.M., Sloane N.J.A.: Upper bounds for modular forms, lattices, and codes. J. Algebra 36, 68–76 (1975).

    Article  MathSciNet  Google Scholar 

  20. Matsuo A.: Norton’s trace formulae for the Griess algebra of a vertex operator algebra with larger symmetry. Commun. Math. Phys. 224, 565–591 (2001).

    Article  MathSciNet  Google Scholar 

  21. Maruoka H., Matsuo A., Shimakura H.: Classification of vertex operator algebras of class \(S^4\) with minimal conformal weight one. J. Math. Soc. Jpn. 68(4), 1369–1388 (2016).

    Article  Google Scholar 

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

  23. Miezaki T.: On a generalization of spherical designs. Discret. Math. 313(4), 375–380 (2013).

    Article  MathSciNet  Google Scholar 

  24. Miezaki T., Munemasa A., Nakasora H.: A note on Assmus-Mattson theorems. Des. Codes Cryptogr. (to appear)

  25. 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(1), 37–46 (2016).

    Article  MathSciNet  Google Scholar 

  26. Pache C.: Shells of selfdual lattices viewed as spherical designs. Int. J. Algebra Comput. 5, 1085–1127 (2005).

    Article  MathSciNet  Google Scholar 

  27. Rains E., Sloane N.J.A.: Self-dual codes. In: Pless V.S., Huffman W.C. (eds.) Handbook of Coding Theory, pp. 177–294. Elsevier, Amsterdam (1998).

    Google Scholar 

  28. Serre J.-P.: Sur la lacunarité des puissances de \(\eta \). Glasgow Math. J. 27, 203–221 (1985).

    Article  MathSciNet  Google Scholar 

  29. Shimakura H.: Private communications (2020).

  30. Venkov B.B.: Even unimodular extremal lattices (Russian), Algebraic geometry and its applications. Trudy Mat. Inst. Steklov. 165, 43–48 (1984).

    MathSciNet  MATH  Google Scholar 

  31. translation in Proc: Steklov Inst. Math. 165, 47–52 (1985).

    Google Scholar 

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

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Acknowledgements

The author would like to thank Prof. Hiroki Shimakura, for his helpful discussions and contributions to this research. The author would also like to thank the anonymous reviewers for their beneficial comments on an earlier version of the manuscript. This work was supported by JSPS KAKENHI (18K03217).

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Correspondence to Tsuyoshi Miezaki.

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Communicated by L. Teirlinck.

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Miezaki, T. Design-theoretic analogies between codes, lattices, and vertex operator algebras. Des. Codes Cryptogr. 89, 763–780 (2021). https://doi.org/10.1007/s10623-021-00842-2

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