New bounds for equiangular lines

Abstract

Abstract. A set of lines in R n is called equiangular if the angle between each pair of lines is the same. We address the question of determining the maximum size of equiangular line sets in R n , using semidefinite programming to improve the upper bounds on this quantity. Improvements are obtained in dimensions 24 ≤ n ≤ 136. In particular, we show that the maximum number of equiangular lines in R n is 276 for all 24 ≤ n ≤ 41 and is 344 for n = 43. This provides a partial resolution of the conjecture set forth by Lemmens and Seidel (1973).

New Upper Bounds for Equiangular Lines by Pillar Decomposition

“…Table 1 presents the currently known s(r) for dimensions 2 ≤ r ≤ 43. An attractive direction Table 1: Maximum number of equiangular lines for small dimensions (Lemmens and Seidel, 1973;Waldron, 2009;Barg and Yu, 2014;Greaves et al, 2016;Yu, 2015;Azarija and Marc, 2016;Szöllősi, 2017;Greaves, 2018;Greaves and Yatsyna, 2018) of research is to develop a general method to compute s(r) or a bound on s(r) for any r ≥ 44. So far, for any r (r ≥ 44), we only know (Greaves et al, 2016, Corollary 2.8) 32r 2 + 328r + 296 1089 ≤ s(r) ≤ r(r + 1) 2 .…”

“…The last sentence in (Neumaier, 1989) says this requires substantially stronger techniques. The good news is that relative bounds for general α can be computed by semidefinite programming (SDP) (Barg and Yu, 2014). The best known non-trivial relative bounds and upper bound of s(r) for 44 ≤ r ≤ 136 that existed before this paper was originally released can be found in (Barg and Yu, 2014, Table 3).…”

Equiangular line systems and switching classes containing regular graphs

“…(The constant α is called the common angle.) Let N (d) denote the maximum cardinality of a system of equiangular lines in dimension d. Determining values of the sequence {N (d)} d∈N is a classical problem that has received much attention [11,16,12] and recently [3,5,8,9,13] there have been some improvements to the upper bounds for N (d) for various values of d. One contribution of this article is to improve the upper bound for N (18) showing that N (18) 60. Furthermore, we show that certain Seidel matrices corresponding to systems of 60 equiangular lines in R 18 each must contain in their switching classes a regular graph having four distinct eigenvalues (see Remark 5.12 below).…”

“…For tables of bounds for equiangular line systems in larger dimensions we refer the reader to Barg and Yu [3].…”

Equiangular Subspaces in Euclidean Spaces

“…The question of determining the maximum size N (n) of a set of equiangular lines in R n has a long history going back 70 years. It is considered to be one of the founding problems of algebraic graph theory, see [2,3,8,14,16,18] and references for more information. It is known that N (n) grows quadratically with n. The upper bound N (n) ≤ n + 1 2 (1) was proved by Gerzon (see [16]) and de Caen [5] gave a (quite nontrivial) construction showing N (n) ≥ 2 9 (n + 1) 2 (2) for all n of the form 3 · 2 2t−1 − 1 where t ∈ N. It is therefore natural and interesting to study analogous questions for k-dimensional subspaces.…”