A002853 Maximal size of a set of equiangular lines in n dimensions. (Formerly M2514 N0994)

1, 3, 6, 6, 10, 16, 28, 28, 28, 28, 28, 28, 28, 28, 36, 40, 48

Offset

1,2

Comments

The sequence continues: 57 <= a(18) <= 60, 72 <= a(19) <= 74, 90 <= a(20) <= 94, a(21) = 126, a(22) = 176, a(23) = ... = a(41) = 276, 276 <= a(42) <= 288, a(43) = 344.

Seidel (1995) claimed, without proof, that a(14) = 28. This was not known at the time. See Greaves, Koolen, Munemasa, Szollosi, (2016). - Ferenc Szollosi, Aug 31 2015

a(14) is now known to be 28 (see Greaves et al. (2020)). - N. J. A. Sloane, Feb 21 2020

References

W. W. R. Ball and H. S. M. Coxeter, "Mathematical Recreations and Essays," 13th Ed. Dover, p. 307.

F. Buekenhout, ed., Handbook of Incidence Geometry, 1995, p. 884.

Greaves, G., Koolen, J. H., Munemasa, A., & Szöllősi, F. (2016). Equiangular lines in Euclidean spaces. Journal of Combinatorial Theory, Series A, 138, 208-235.

Lin, Yen-Chi Roger, and Wei-Hsuan Yu. "Equiangular lines and the Lemmens-Seidel conjecture." Discrete Mathematics 343.2 (2020): 111667.

Lin, Yen-Chi Roger, and Wei-Hsuan Yu. "Saturated configuration and new large construction of equiangular lines", Linear Algebra Appl., 588, 272-281, 2020.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Table of n, a(n) for n=1..17.

I. Ballar, F. Draxler, P. Keevash, and B. Sudakov, Equiangular Lines and Spherical Codes in Euclidean Space, arxiv preprint arxiv:1606.06620 [math.HO], 2016.

A. Barg and W.-H. Yu, New bounds for equiangular lines, arXiv:1311.3219 [math.MG], 2014.

A. Barg and W.-H. Yu, New bounds for equiangular lines, Contemporary Math. vol. 625, 2014, pp. 111--121.

David de Laat, Fabrício Caluza Machado, Fernando Mário de Oliveira Filho, and Frank Vallentin, k-point semidefinite programming bounds for equiangular lines, arXiv:1812.06045 [math.OC], 2018.

Christopher A. Fuchs, Maxim Olchanyi, and Matthew B. Weiss, Quantum mechanics? It's all fun and games until someone loses an i, arXiv:2206.15343 [quant-ph], 2022.

G. Greaves, Equiangular line systems and switching classes containing regular graphs, Linear Algebra Appl. 536, pp. 31--51 (2018).

Gary R. W. Greaves and Jeven Syatriadi, Real equiangular lines in dimension 18 and the Jacobi identity for complementary subgraphs, J. Comb. Theor. A (2024) Vol. 201, 105812. See p. 3.

Gary R. W. Greaves, Jeven Syatriadi, and Pavlo Yatsyna, Equiangular lines in low dimensional Euclidean spaces, arXiv:2002.08085 [math.CO], 2020.

Gary R. W. Greaves, Jeven Syatriadi, and Pavlo Yatsyna, Equiangular lines in Euclidean spaces: dimensions 17 and 18, arXiv:2104.04330 [math.CO], 2021.

G. Greaves, J. H. Koolen, A. Munemasa, and F. Szollosi, Equiangular lines in Euclidean spaces, J. Combin. Theory Ser. A 138, pp. 208--235 (2016).

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

K. Hartnett, A New Path To Equal Angle Lines, Quanta Magazine, Apr 11, 2017.

P. W. H. Lemmens and J. J. Seidel, Equiangular lines, J. Algebra, 24 (1973), 494-512.

Yen-Chi Roger Lin, and Wei-Hsuan Yu, Equiangular lines and the Lemmens-Seidel conjecture, arXiv:1807.06249 [math.CO], 2019.

G. McConnell, Some non-standard ways to generate SIC-POVMs in dimensions 2 and 3, arXiv preprint arXiv:1402.7330 [quant-ph], 2014. See Abstract.

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

Blake C. Stacey, Geometric and Information-Theoretic Properties of the Hoggar Lines, arXiv preprint arXiv:1609.03075 [quant-ph], 2016.

Blake C. Stacey, Quantum Theory as Symmetry Broken by Vitality, arXiv:1907.02432 [quant-ph], 2019.

F. Szollosi, A Remark on a Construction of D.S. Asche, Discrete Comput. Geom. (2017).

Crossrefs

Cf. A332546.

Sequence in context: A316140 A147849 A332546 * A278807 A340620 A184137

Adjacent sequences: A002850 A002851 A002852 * A002854 A002855 A002856

Keyword

nonn,nice,hard,more

Author

N. J. A. Sloane

Extensions

Terms above a(14) removed by Ferenc Szollosi, Aug 31 2015

Updates to a(14), a(15), a(16), a(19), a(20) added from Greaves et al. (2020) by N. J. A. Sloane, Feb 21 2020. Thanks to Yen-Chi Roger Lin for telling us about this paper.

Updates to a(17) and a(18) added from Greaves et al. (2021) by Gary R. W. Greaves, Jul 10 2021

Status

approved