A new relative bound for equiangular lines and nonexistence of tight spherical designs of harmonic index 4

Equiangular lines in Euclidean spaces

“…In particular, there have emerged improvements to the lower bounds for N (18) [24,36] and improvements to the upper bounds for N (d) where d = 14, 16, 17, 18, 19, 20 [1, 2, 14, 15, 16]. There have also been various recent improvements to upper bounds for N (d) for d 24 using semidefinite programming, see [12,21,22,28,38].…”

“…In particular, there have emerged improvements to the lower bounds for N (18) [24,36] and improvements to the upper bounds for N (d) where d = 14, 16, 17, 18, 19, 20 [1, 2, 14, 15, 16]. There have also been various recent improvements to upper bounds for N (d) for d 24 using semidefinite programming, see [12,21,22,28,38].The asymptotic behaviour of N (d) is quadratic in d with a general upper bound of d(d+1)/2 [23, Theorem 3.5] and a general lower bound of (32d 2 + 328d + 29)/1089 [15, Corollary 2.8]. 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.…”

k-Point semidefinite programming bounds for equiangular lines

“…5.1 when k = 3 (except for an ad hoc 2 × 2 matrix). In the other papers [5,27,41,48] where this semidefinite program is considered, a primal version is given instead, which is less convenient from the perspective of rigorous verification of bounds.…”

Classification of Spherical $2$-distance $\{4,2,1\}$-designs by Solving Diophantine Equations

“…The study of spherical Tdesigns started from Bannai-Okuda-Tagami [6]. Later Okuda-Yu [17] proved the nonexistence of tight spherical {4}-designs. Some further discussion of spherical T -designs can be found in [20].…”