%I M2514 N0994 #117 Sep 12 2023 09:47:14
%S 1,3,6,6,10,16,28,28,28,28,28,28,28,28,36,40,48
%N Maximal size of a set of equiangular lines in n dimensions.
%C 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.
%C 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
%C a(14) is now known to be 28 (see Greaves et al. (2020)). - _N. J. A. Sloane_, Feb 21 2020
%D W. W. R. Ball and H. S. M. Coxeter, "Mathematical Recreations and Essays," 13th Ed. Dover, p. 307.
%D F. Buekenhout, ed., Handbook of Incidence Geometry, 1995, p. 884.
%D Greaves, G., Koolen, J. H., Munemasa, A., & Szöllősi, F. (2016). Equiangular lines in Euclidean spaces. Journal of Combinatorial Theory, Series A, 138, 208-235.
%D Lin, Yen-Chi Roger, and Wei-Hsuan Yu. "Equiangular lines and the Lemmens-Seidel conjecture." Discrete Mathematics 343.2 (2020): 111667.
%D Lin, Yen-Chi Roger, and Wei-Hsuan Yu. "Saturated configuration and new large construction of equiangular lines", Linear Algebra Appl., 588, 272-281, 2020.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H I. Ballar, F. Draxler, P. Keevash, and B. Sudakov, <a href="https://arxiv.org/abs/1606.06620">Equiangular Lines and Spherical Codes in Euclidean Space</a>, arxiv preprint arxiv:1606.06620 [math.HO], 2016.
%H A. Barg and W.-H. Yu, <a href="http://arxiv.org/abs/1311.3219">New bounds for equiangular lines</a>, arXiv:1311.3219 [math.MG], 2014.
%H A. Barg and W.-H. Yu, <a href="http://www.ams.org/books/conm/625/">New bounds for equiangular lines</a>, Contemporary Math. vol. 625, 2014, pp. 111--121.
%H David de Laat, Fabrício Caluza Machado, Fernando Mário de Oliveira Filho, and Frank Vallentin, <a href="https://arxiv.org/abs/1812.06045">k-point semidefinite programming bounds for equiangular lines</a>, arXiv:1812.06045 [math.OC], 2018.
%H Christopher A. Fuchs, Maxim Olchanyi, and Matthew B. Weiss, <a href="https://arxiv.org/abs/2206.15343">Quantum mechanics? It's all fun and games until someone loses an i</a>, arXiv:2206.15343 [quant-ph], 2022.
%H G. Greaves, <a href="https://doi.org/10.1016/j.laa.2017.09.008">Equiangular line systems and switching classes containing regular graphs</a>, Linear Algebra Appl. 536, pp. 31--51 (2018).
%H Gary R. W. Greaves and Jeven Syatriadi, <a href="https://doi.org/10.1016/j.jcta.2023.105812">Real equiangular lines in dimension 18 and the Jacobi identity for complementary subgraphs</a>, J. Comb. Theor. A (2024) Vol. 201, 105812. See p. 3.
%H Gary R. W. Greaves, Jeven Syatriadi, and Pavlo Yatsyna, <a href="https://arxiv.org/abs/2002.08085">Equiangular lines in low dimensional Euclidean spaces</a>, arXiv:2002.08085 [math.CO], 2020.
%H Gary R. W. Greaves, Jeven Syatriadi, and Pavlo Yatsyna, <a href="https://arxiv.org/abs/2104.04330">Equiangular lines in Euclidean spaces: dimensions 17 and 18</a>, arXiv:2104.04330 [math.CO], 2021.
%H G. Greaves, J. H. Koolen, A. Munemasa, and F. Szollosi, <a href="https://doi.org/10.1016/j.jcta.2015.09.008">Equiangular lines in Euclidean spaces</a>, J. Combin. Theory Ser. A 138, pp. 208--235 (2016).
%H G. Greaves and P. Yatsyna, <a href="https://doi.org/10.1090/mcom/3433">On equiangular lines in 17 dimensions and the characteristic polynomial of a Seidel matrix</a>, Math. Comp. 88 (2019), pp. 3041--3061.
%H K. Hartnett, <a href="https://www.quantamagazine.org/20170411-equiangular-lines-proof/">A New Path To Equal Angle Lines</a>, Quanta Magazine, Apr 11, 2017.
%H P. W. H. Lemmens and J. J. Seidel, <a href="http://dx.doi.org/10.1016/0021-8693(73)90123-3">Equiangular lines</a>, J. Algebra, 24 (1973), 494-512.
%H Yen-Chi Roger Lin, and Wei-Hsuan Yu, <a href="https://arxiv.org/abs/1807.06249">Equiangular lines and the Lemmens-Seidel conjecture</a>, arXiv:1807.06249 [math.CO], 2019.
%H G. McConnell, <a href="http://arxiv.org/abs/1402.7330">Some non-standard ways to generate SIC-POVMs in dimensions 2 and 3</a>, arXiv preprint arXiv:1402.7330 [quant-ph], 2014. See Abstract.
%H J. J. Seidel, <a href="https://doi.org/10.1016/B978-044488355-1/50017-7">Discrete non-Euclidean geometry</a>, in Buekenhout (ed.), Handbook of Incidence Geometry, Elsevier, Amsterdam, The Nederlands (1995).
%H Blake C. Stacey, <a href="https://arxiv.org/abs/1609.03075">Geometric and Information-Theoretic Properties of the Hoggar Lines</a>, arXiv preprint arXiv:1609.03075 [quant-ph], 2016.
%H Blake C. Stacey, <a href="https://arxiv.org/abs/1907.02432">Quantum Theory as Symmetry Broken by Vitality</a>, arXiv:1907.02432 [quant-ph], 2019.
%H F. Szollosi, <a href="https://doi.org/10.1007/s00454-017-9933-4">A Remark on a Construction of D.S. Asche</a>, Discrete Comput. Geom. (2017).
%Y Cf. A332546.
%K nonn,nice,hard,more
%O 1,2
%A _N. J. A. Sloane_
%E Terms above a(14) removed by _Ferenc Szollosi_, Aug 31 2015
%E 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.
%E Updates to a(17) and a(18) added from Greaves et al. (2021) by _Gary R. W. Greaves_, Jul 10 2021