Equiangular lines and spherical codes in Euclidean space

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 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 θ and sufficiently large n there are at most 2n − 2 lines in R n with common angle θ. Moreover, this bound is achieved if and only if θ = arccos . Indeed, we show that for all θ = arccos 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 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.

Equiangular lines in Euclidean spaces

“…One can also consider the related problem of, for fixed α ∈ (0, 1), finding N α (d), the maximum number of lines in R d through the origin with pairwise angle arccos α. Motivated by a conjecture of Bukh [4], the asymptotic behaviour of N α (d) was recently shown to be linear in d [3,20].In Table 1 below, we give the currently known (including the improvements from this paper) values or lower and upper bounds for N (d) for d at most 23.…”

Open problems in the spectral theory of signed graphs

“…Seidel matrices were introduced by Van Lint and Seidel [62] and studied by many people due to their interesting properties and connections to equiangular lines, two-graphs, strongly regular graphs, mutually unbiased bases and so on (see [10,Section 10.6] and [4,45,73] for example). The connection between Seidel matrices and equiangular lines is perhaps best summarized in [10, p.161]:…”

“…[60,Theorem 3.4]) proved that if N α (d) ≥ 2d, then 1/α is an odd integer. Bukh [12] proved that N α (d) ≤ c α d, where c α is a constant depending only on α. Balla, Dräxler, Keevash and Sudakov [4] improved this bound and showed that for d sufficiently large and α = 1/3, When 1/α is not a totally real algebraic integer, then N α (d) = d. Jiang and Polyanskii [56] studied the set T = {α | α ∈ (0, 1), lim sup d→∞ N α (d)/d > 1} and showed that the closure of T contains the closed interval [0, 1/ √ 5 + 2] using results of Shearer [75] on the spectral radius of unsigned graphs.…”

“…Glazyrin and Yu [39] obtained a general bound N α (d) ≤ 2α 2 /3 + 4/7 d + 2 for all n. Bukh[12] and also, Balla, Dräxler, Keevash and Sudakov[4] conjecture the following.…”

Complete multipartite graphs that are determined, up to switching, by their Seidel spectrum

“…A Seidel switching with respect to U transforms G to a graph H by deleting the edges between U and W and adding an edge between vertices u ∈ U and w ∈ W if (u, w) / ∈ E(G). For more details on Seidel matrices and related topics, see [15,11,12,1] and the references there. Seidel switching is an equivalence relation and we say that G and H are switching equivalent.…”