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Inventiones mathematicae

, Volume 211, Issue 1, pp 179–212 | Cite as

Equiangular lines and spherical codes in Euclidean space

  • Igor Balla
  • Felix Dräxler
  • Peter Keevash
  • Benny Sudakov
Article

Abstract

A family of lines through the origin in Euclidean space is called equiangular if any pair of lines defines the same angle. The problem of estimating the maximum cardinality of such a family in \(\mathbb {R}^n\) was extensively studied for the last 70 years. Motivated by a question of Lemmens and Seidel from 1973, in this paper we prove that for every fixed angle \(\theta \) and sufficiently large \(n\) there are at most \(2n-2\) lines in \(\mathbb {R}^n\) with common angle \(\theta \). Moreover, this bound is achieved if and only if \(\theta = \arccos \frac{1}{3}\). Indeed, we show that for all \(\theta \ne \arccos {\frac{1}{3}}\) and and sufficiently large n, the number of equiangular lines is at most 1.93n. We also show that for any set of k fixed angles, one can find at most \(O(n^k)\) lines in \(\mathbb {R}^n\) having these angles. This bound, conjectured by Bukh, substantially improves the estimate of Delsarte, Goethals and Seidel from 1975. Various extensions of these results to the more general setting of spherical codes will be discussed as well.

Notes

Acknowledgements

We would like to thank Boris Bukh and the referee for useful comments.

References

  1. 1.
    Alon, N.: Perturbed identity matrices have high rank: proof and applications. Combin. Probab. Comput. 18, 3–15 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Barg, A., Yu, W.-H.: New bounds for equiangular lines. Contemp. Math. 625, 111–121 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bellman, R.: Introduction to Matrix Analysis, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia (1997)zbMATHGoogle Scholar
  4. 4.
    Blumenthal, L.M.: Theory and Applications of Distance Geometry. Clarendon Press, Oxford (1953)zbMATHGoogle Scholar
  5. 5.
    Bukh, B.: Bounds on equiangular lines and on related spherical codes. SIAM J. Discrete Math. 30, 549–554 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    de Caen, D.: Large equiangular sets of lines in Euclidean space. Electron. J. Combin. 7 (Research paper 55) (2000)Google Scholar
  7. 7.
    Coxeter, H.S.M.: Regular Polytopes, 3rd edn. Dover Publications, New York (1973)zbMATHGoogle Scholar
  8. 8.
    Delsarte, P., Goethals, J.M., Seidel, J.J.: Bounds for systems of lines, and Jacobi polynomials. Philips Res. Rep. 30, 91–105 (1975)zbMATHGoogle Scholar
  9. 9.
    Delsarte, P., Goethals, J.M., Seidel, J.J.: Spherical codes and designs. Geometriae Dedicata 6, 363–388 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dvir, Z., Saraf, S., Wigderson, A.: Improved rank bounds for design matrices and a new proof of Kelly’s theorem. Forum Math. Sigma 2, e4 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Erdős, P., Szekeres, G.: A combinatorial problem in geometry. Compositio Math. 2, 463–470 (1935)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Greaves, G., Koolen, J.H., Munemasa, A., Szöllősi, F.: Equiangular lines in Euclidean spaces. J. Combin. Theory Ser. A 138, 208–235 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Godsil, C., Royle, G.: Algebraic Graph Theory. Springer, New York (2001)CrossRefzbMATHGoogle Scholar
  14. 14.
    Haantjes, J.: Equilateral point-sets in elliptic two- and three-dimensional spaces. Nieuw Arch. Wiskunde 22, 355–362 (1948)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Haantjes, J., Seidel, J.J.: The congruence order of the elliptic plane. Indag. Math. 9, 403–405 (1947)zbMATHGoogle Scholar
  16. 16.
    Heath, R.W., Strohmer, T.: Grassmannian frames with applications to coding and communication. Appl. Comput. Harmon. Anal. 14, 257–275 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Jedwab, J., Wiebe, A.: Large sets of complex and real equiangular lines. J. Combin. Theory Ser. A 134, 98–102 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Koornwinder, T.H.: A note on the absolute bound for systems of lines. Indag. Math. (Proceedings) 79, 152–153 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lemmens, P.W.H., Seidel, J.J.: Equiangular lines. J. Algebra 24, 494–512 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    van Lint, J.H., Seidel, J.J.: Equilateral point sets in elliptic geometry. Indag. Math. 28, 335–348 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Lovász, L.: Combinatorial Problems and Exercises, 2nd edn. AMS Chelsea Publishing, Rhode Island (2014)zbMATHGoogle Scholar
  22. 22.
    Milman, V.D., Schechtman, G.: Asymptotic Theory of Finite Dimensional Normed Spaces. Springer, New York (1986)zbMATHGoogle Scholar
  23. 23.
    Neumaier, A.: Graph representations, two-distance sets, and equiangular lines. Linear Algebra Appl. 114, 141–156 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Nilli, A.: On the second eigenvalue of a graph. Discrete Math. 91(2), 207–210 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Rankin, R.A.: The Closest Packing of Spherical Caps in \(n\) Dimensions, vol. 2. Cambridge University Press, Cambridge, pp. 139–144 (1955)Google Scholar
  26. 26.
    Schnirelmann, L.G.: On the additive properties of numbers. Proc. Don Polytech. Inst. Novocherkassk 14, 3–27 (1930)Google Scholar
  27. 27.
    Seidel, J.J.: Strongly regular graphs of \(L_2\)-type and of triangular type. Indag. Math. 29, 188–196 (1967)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Igor Balla
    • 1
  • Felix Dräxler
    • 1
  • Peter Keevash
    • 2
  • Benny Sudakov
    • 1
  1. 1.Department of MathematicsETHZurichSwitzerland
  2. 2.Mathematical InstituteUniversity of OxfordOxfordUK

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