Skip to main content
SpringerLink
Log in

Codes, Cubes, and Graphical Designs

  • Published:
Journal of Fourier Analysis and Applications Aims and scope Submit manuscript

Abstract

Graphical designs are an extension of spherical designs to functions on graphs. We connect linear codes to graphical designs on cube graphs, and show that the Hamming code in particular is a highly effective graphical design. We show that even in highly structured graphs, graphical designs are distinct from the related concepts of extremal designs, maximum stable sets in distance graphs, and t-designs on association schemes.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  1. Alon, N., Milman, V.D.: \(\lambda _1\), Isoperimetric inequalities for graphs, and superconcentrators. J. Comb. Theory 38(1), 73–88 (1985)

    Article  Google Scholar 

  2. Anis, A., Gadde, A., Ortega, A.: Efficient sampling set selection for bandlimited graph signals using graph spectral proxies. In: IEEE Transactions on Signal Processing a Publication of the IEEE Signal Processing Society, vol. 64(14), pp. 3775–3789 (2016)

  3. Axler, S., Bordon, P., Ramey, W.: Harmonic Function Theory. Springer, New York (1992)

    Book  Google Scholar 

  4. Bondarenko, A., Radchenko, D., Viazovska, M.: Optimal asymptotic bounds for spherical designs. Ann. Math. 178(2), 443–452 (2013)

    Article  MathSciNet  Google Scholar 

  5. Bondarenko, A., Radchenko, D., Viazovska, M.: Well-separated spherical designs. Construct. Approx. 41(1), 93–112 (2015)

    Article  MathSciNet  Google Scholar 

  6. Bose, R.C., Mesner, D.M.: On linear associative algebras corresponding to association schemes of partially balanced designs. Ann. Math. Stat. 30(1), 21–38 (1959). https://doi.org/10.1214/aoms/1177706356

    Article  MathSciNet  MATH  Google Scholar 

  7. Brouwer, A.E., Haemers, W.H.: Spectra of Graphs. Springer, New York (2012)

    Book  Google Scholar 

  8. Brouwer, A.E., Cohen, A.M., Neumaier, A.: Distance-Regular Graphs. Springer, Berlin (1989)

    Book  Google Scholar 

  9. Chung, F.R.K.: Spectral graph theory. Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, Providence, RI (1997)

  10. Delsarte, Ph.: An algebraic approach to the association schemes of coding theory. Philips Res. Rept. Suppl. (1973). https://doi.org/10.1007/978-94-010-1826-5_7

    Article  MathSciNet  MATH  Google Scholar 

  11. Delsarte, Ph., Goethal, J.M., Seidel, J.J.: Spherical codes and designs. Geometr. Ded. 6(3), 363–388 (1977)

    Article  MathSciNet  Google Scholar 

  12. Gariboldi, B., Gigante, G.: Optimal asymptotic bounds for designs on manifolds (2018). arXiv: 1811.12676 [math.AP]

  13. Golubev, K.: Graphical designs and extremal combinatorics. Linear Algebra Appl. 604, 490 (2020)

    Article  MathSciNet  Google Scholar 

  14. Haemers, W.H.: Hoffman’s ratio bound (2021). arXiv: 2102.05529 [math.CO]

  15. Hamming, R.W.: Error detecting and error correcting codes. Bell Syst. Techn. J. 29(2), 147–160 (1950)

    Article  MathSciNet  Google Scholar 

  16. Hoffman, A.J.: On eigenvalues and colorings of graphs. In: Harris, B. (ed.) Graph Theory and Its Applications. Routledge, London (1970)

    Google Scholar 

  17. Lebedev, V.I.: Quadratures on the sphere. Zh. Vchisl. Mat. Mat. Fiz. i. 16(2), 293–306 (1976)

    MathSciNet  Google Scholar 

  18. Linderman, G.C., Steinerberger, S.: Numerical integration on graphs: where to sample and how to weigh. Math. Comput. 89(324), 1933–1952 (2020)

    Article  MathSciNet  Google Scholar 

  19. MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error Correcting Codes. North Holland, Amsterdam (1977)

    MATH  Google Scholar 

  20. MATLAB. version 9.9.0 (R2020b). The MathWorks Inc., Natick (2020)

  21. Mohlenkamp, M.J.: A User’s Guide to Spherical Harmonics. http://www.ohiouniversityfaculty.com/mohlenka/research/uguide.pdf (version: 10.18.2016)

  22. Pesenson, I.Z.: Sampling by averages and average splines on Dirichlet spaces and on combinatorial graphs (2019). arXiv: 1901.08726

  23. Pesenson, I.Z.: A sampling theorem on homogeneous manifolds. Trans. Am. Math. Soc. 352(9), 4257–4269 (2000)

    Article  MathSciNet  Google Scholar 

  24. Pesenson, I.Z.: Sampling in Paley–Wiener spaces on combinatorial graphs. Trans. Am. Math. Soc. 360(10), 5603–5627 (2008)

    Article  MathSciNet  Google Scholar 

  25. Sagan, B.E.: The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, vol. 203, 2nd edn. Springer, New York (2001)

    Book  Google Scholar 

  26. Sobolev, S.: Cubature formulas on the sphere which are invariant under transformations of finite rotation groups. Dokl. Akad. Nauk SSSR 146, 310–313 (1962)

    MathSciNet  Google Scholar 

  27. Steinerberger, S.: Spectral Limitations of Quadrature Rules and Generalized Spherical Designs. In: IMRN (2019). https://arxiv.org/abs/1708.08736 (accepted)

  28. Steinerberger, S.: Generalized designs on graphs: sampling, spectra, symmetries. J. Graph Theory 93(2), 253–267 (2020)

    Article  MathSciNet  Google Scholar 

  29. Stinson, D.R.: Combinatorial Designs: Construction and Analysis. Springer, New York (2004)

    MATH  Google Scholar 

  30. Suzuki, H.: Association schemes with multiple Q-polynomial structures. J. Algebraic Comb. 7, 181–196 (1998)

    Article  MathSciNet  Google Scholar 

  31. Tanaka, Y., et al.: Sampling signals on graphs: from theory to applications. IEEE Signal Process. Mag. 37(6), 14–30 (2020). https://doi.org/10.1109/msp.2020.3016908

    Article  Google Scholar 

  32. Tanner, R.M.: Explicit concentrators from generalized n-gons. SIAM J. Algebra. Discret. Methods 5(3), 287–293 (1984)

    Article  MathSciNet  Google Scholar 

  33. Tsitsvero, M., Barbarossa, S., Di Lorenzo, P.: Signals on graphs: uncertainty principle and sampling. In: IEEE Transactions on Signal Processing a Publication of the IEEE Signal Processing Society, vol. 64(18), pp. 4845–4860 (2016)

  34. Vallentin, F.: Semidefinite programming bounds for error-correcting codes. In: CoRR abs/1902.01253 (2019). arXiv: 1902.01253

  35. Weyl, H.: Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung). Math. Ann. 71(4), 441–479 (1912)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author would like to thank Rekha Thomas and Stefan Steinerberger for their guidance and help preparing this manuscript, Shahar Kovalsky for sharing code that assisted in computations, and Ferdinand Ihringer for noting some missing details in Sect. 6. Figures 1 and 3a–c are due to Stefan Steinerberger. Figure 3d is used with permission, copyright \(\copyright \)2015, PRISM Climate Group, Oregon State University, http://prism.oregonstate.edu/normals/; retrieved 3 Dec 2020. Computations were done in [20].

Funding

Research partially supported by the U.S. National Science Foundation grant DMS-1719538.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Washington, Seattle, WA, 98105, USA

    Catherine Babecki

Authors
  1. Catherine Babecki
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Catherine Babecki.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Babecki, C. Codes, Cubes, and Graphical Designs. J Fourier Anal Appl 27, 81 (2021). https://doi.org/10.1007/s00041-021-09852-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00041-021-09852-z

Keywords

Mathematics Subject Classification

Search

Navigation