Skip to main content
SpringerLink
Log in

Equiangular Lines in Low Dimensional Euclidean Spaces

  • Original paper
  • Published:
Combinatorica Aims and scope Submit manuscript

Abstract

We show that the maximum cardinality of an equiangular line system in 14 and 16 dimensions is 28 and 40, respectively, thereby solving a longstanding open problem. We also improve the upper bounds on the cardinality of equiangular line systems in 19 and 20 dimensions to 74 and 94, respectively.

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. J. Azarija and T. Marc: There is no (75, 32, 10, 16) strongly regular graph, Linear Algebra Appl. 557 (2018), 62–83.

    Article  MathSciNet  Google Scholar 

  2. J. Azarija and T. Marc: There is no (95, 40, 12, 20) strongly regular graph, J. Combin. Des. 28 (2020), 294–306.

    Article  MathSciNet  Google Scholar 

  3. I. Balla, F. Draxler, P. Keevash and B. Sudakov: Equiangular lines and spherical codes in Euclidean space, Invent. Math. 211 (2018), 179–212.

    Article  MathSciNet  Google Scholar 

  4. B. Bukh: Bounds on equiangular lines and on related spherical codes, SIAM J. Discrete Math. 30 (2016), 549–554.

    Article  MathSciNet  Google Scholar 

  5. A.-L. Cauchy: Sur l’équation à l’aide de laquelle on détermine les inégalités séculaires des mouvements des planètes, in: Œuvres complètes, IIième Série, Gauthier-Villars, Paris, 1829.

    Google Scholar 

  6. D. Cvetković, P. Rowlinson and S. Simić: Eigenspaces of graphs, Encyclopedia of Mathematics and its Applications, vol. 66, Cambridge University Press, Cambridge, 1997.

    Book  Google Scholar 

  7. P. B. Denton, S. J. Parke, T. Tao and X. Zhang: Eigenvectors from Eigenvalues: a survey of a basic identity in linear algebra, arXiv:1908.03795, 2019.

  8. J. Farkas: Theorie der einfachen Ungleichungen, Journal für die reine und angewandte Mathematik 124 (1902), 1–27.

    MathSciNet  MATH  Google Scholar 

  9. S. Fisk: A very short proof of Cauchy’s interlace theorem for eigenvalues of Hermitian matrices, Amer. Math. Monthly 112 (2005), 118.

    Google Scholar 

  10. C. D. Godsil and B. D. McKay: Spectral conditions for the reconstructibility of a graph, J. Combin. Theory Ser. B 30 (1981), 285–289.

    Article  MathSciNet  Google Scholar 

  11. A. Glazyrin and W.-H. Yu: Upper bounds for s-distance sets and equiangular lines, Adv. Math. 330 (2018), 810–833.

    Article  MathSciNet  Google Scholar 

  12. G. R. W. Greaves: SageMath notebook containing computations from the paper “Equiangular lines in low dimensional Euclidean spaces”, https://github.com/grwgrvs/interlacing-characteristic-polynomials/.

  13. G. R. W. Greaves: Equiangular line systems and switching classes containing regular graphs, Linear Algebra Appl. 536 (2018), 31–51.

    Article  MathSciNet  Google Scholar 

  14. G. R. W. Greaves, J. Koolen, A. Munemasa and F. Szöllősi: Equiangular lines in Euclidean spaces, J. Combin. Theory Ser. A 138 (2016), 208–235.

    Article  MathSciNet  Google Scholar 

  15. G. R. W. Greaves and P. Yatsyna: On equiangular lines in 17 dimensions and the characteristic polynomial of a Seidel matrix, Math. Comp. 88 (2019), 3041–3061.

    Article  MathSciNet  Google Scholar 

  16. J. Haantjes: Equilateral point-sets in elliptic two- and three-dimensional spaces, Nieuw Arch. Wiskd. 22 (1948), 355–362.

    MathSciNet  MATH  Google Scholar 

  17. R. W. Heath and T. Strohmer: Grassmannian frames with applications to coding and communication, Appl. Comput. Harmon. Anal. 14(3) (2003), 257–275.

    Article  MathSciNet  Google Scholar 

  18. S.-G. Hwang: Cauchy’s interlace theorem for eigenvalues of Hermitian matrices, Amer. Math. Monthly 111 (2004), 157–159.

    Article  MathSciNet  Google Scholar 

  19. Z. Jiang, J. Tidor, Y. Yao, S. Zhang and Y. Zhao: Equiangular lines with a fixed angle, arXiv:1907.12466, 2019.

  20. E. King and X. Tang: New upper bounds for equiangular lines by pillar decomposition, SIAM J. Discrete Math. 33 (2019), 2479–2508.

    Article  MathSciNet  Google Scholar 

  21. D. de Laat, F. C. Machado, F. M. de Oliveira Filho and F. Vallentin: k-point semidefinite programming bounds for equiangular lines, Math. Program. (2021).

  22. P. W. H. Lemmens and J. J. Seidel: Equiangular lines, J. Algebra 24(3) (1973), 494–512.

    Article  MathSciNet  Google Scholar 

  23. Y.-C. Lin and W.-H. Yu: Saturated configuration and new large construction of equiangular lines, Linear Algebra Appl. 588 (2020), 272–281.

    Article  MathSciNet  Google Scholar 

  24. J. H. van Lint and J. J. Seidel: Equilateral point sets in elliptic geometry, Indag. Math. 28 (1966), 335–348.

    Article  MathSciNet  Google Scholar 

  25. Wolfram Research, Inc., Mathematica, Version Number 10.0, Champaign, IL (2014).

  26. J. F. McKee and C. J. Smyth: Salem numbers of trace −2 and traces of totally positive algebraic integers, Algorithmic number theory, 327–337, Lecture Notes in Comput. Sci., 3076, Springer, Berlin, 2004.

    Chapter  Google Scholar 

  27. T. Okuda and W.-H. Yu: A new relative bound for equiangular lines and nonexistence of tight spherical designs of harmonic index 4, European J. Combin. 53 (2016), 96–103.

    Article  MathSciNet  Google Scholar 

  28. The PARI Group, PARI/GP version 2.10, Univ. Bordeaux, http://pari.math.u-bordeaux.fr/.

  29. R. C. Thompson: Principal submatrices V: Some results concerning principal submatrices of arbitrary matrices, J. Res. Nat. Bur. Standards Sect. B 72 (1968), 115–125.

    Article  MathSciNet  Google Scholar 

  30. J. J. Seidel: Graphs and two-graphs, in: Proc. 5th Southeastern Conf. on Combinatorics, Graph Theory, and Computing, Utilitas Mathematica Publishing Inc., Winnipeg, Canada, 1974.

    MATH  Google Scholar 

  31. J. J. Seidel: Discrete non-Euclidean geometry, in: Buekenhout (ed.), Handbook of Incidence Geometry, Elsevier Science, Amsterdam, The Netherlands (1995).

    Google Scholar 

  32. N. J. A. Sloane (editor): The On-Line Encyclopedia of Integer Sequences, Published electronically at https://oeis.org/.

  33. W. A. Stein et al.: Sage Mathematics Software (Version 8.8), The Sage Development Team, 2010, http://www.sagemath.org/.

  34. F. Szöllősi: A remark on a construction of D. S. Asche, Discrete Comput. Geom. 61 (2019), 120–122.

    Article  MathSciNet  Google Scholar 

  35. D. E. Taylor: Regular 2-graphs, Proc. London Math. Soc. 35 (1977), 257–274.

    Article  MathSciNet  Google Scholar 

  36. W.-H. Yu: New bounds for equiangular lines and spherical two-distance sets, SIAM J. Discrete Math. 31 (2017), 908–917.

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgement

GRWG was supported by the Singapore Ministry of Education Academic Research Fund (Tier 1); grant numbers: RG29/18 and RG21/20. PY was supported by project PRIMUS/20/SCI/002 from Charles University.

Author information

Authors and Affiliations

  1. Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore, 637371, Singapore

    Gary R. W. Greaves & Jeven Syatriadi

  2. Department of Algebra, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 18600, Praha 8, Czech Republic

    Pavlo Yatsyna

Authors
  1. Gary R. W. Greaves
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jeven Syatriadi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pavlo Yatsyna
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gary R. W. Greaves.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Greaves, G.R.W., Syatriadi, J. & Yatsyna, P. Equiangular Lines in Low Dimensional Euclidean Spaces. Combinatorica 41, 839–872 (2021). https://doi.org/10.1007/s00493-020-4523-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00493-020-4523-0

Mathematics Subject Classification (2010)

Search

Navigation