Equiangular lines and the Lemmens–Seidel conjecture

k-Point semidefinite programming bounds for equiangular lines

“…Recently, Lin and Yu [33] made progress in this conjecture by proving some claims from Lemmens and Seidel [31]. The only case still open is when the code has a set with 4 unit vectors with mutual inner products −1/5 and no such set with 5 unit vectors (up to replacement of some vectors by their antipodes).…”

“…King and Tang [27] improved the pillar decomposition technique and got a better bound for M 1/5 (n) [27,Theorem 7]. Recently, Lin and Yu [33] further improved parts of their argument; by combining [33,Proposition 4.5] with the proof of [27,Theorem 7] we get: Theorem 6.12 (Lin and Yu [33]) If n ≥ 63, then…”

“…6 We also improve the bounds computed by King and Tang [27] for M 1/5 (n) by replacing their theorem [27,Theorem 7] by Theorem 6.12 and by using 6 (G) * to compute better bounds for A(n, {1/13, −5/13}). Lin and Yu [33] observed that A(n, {1/13, −5/13}) ≥ 3n/2 − 3 and therefore there is a limit to the power of this approach: it will never be able to prove Conjecture 6.9 no matter how much we increase k. In general, it is not clear how good the bound k (G) * can be for M a (n) if one allows k to increase; in contrast, de Laat and Vallentin [11,Theorem 2] show that their version of the Lasserre hierarchy for compact topological packing graphs converges to the independence number if enough steps are computed. Whether such a convergence result holds for k (G) * is an open question; in any case, it takes days to compute k (G) * for k = 6, so one can expect that solving the resulting semidefinite programs for k > 6 will be hard in practice.…”

Spectral Proofs of Maximality of Some Seidel Matrices

“…Lin and Yu [9,10] defined a set X of equiangular lines of rank r to be saturated if there is no line l 6 2 X such that the union X [ flg is a set of equiangular lines of rank r. Here, the rank of a set of equiangular lines is the smallest dimension of Euclidean spaces into which these lines are isometrically embedded. By using a computer implementing their algorithm [10, p. 274], they verified in [10, Theorem 1 and the end of Sect.…”

Maximality of Seidel matrices and switching roots of graphs

“…Note that, if a Seidel matrix S has largest eigenvalue λ, then there exist vectors whose Gram matrix equals λI − S. In this case, such vectors span equiangular lines with common angle arccos(1/λ), and the rank of λI − S equals that of these lines. Note that S is maximal if and only if the set of equiangular lines so obtained is saturated in the sense of [8,9]. For example, S := J 4 − I 4 is a Seidel matrix having largest eigenvalue λ = 3, and induces the set of equiangular lines Ru 1 , Ru 2 , Ru 3 and Ru 4 with common angle arccos(1/3), where u 1 := (1, 1, 1, 0, 0, 0) ⊤ / √ 3, u 2 := (−1, 0, 0, 1, 1, 0) ⊤ / √ 3, u 3 := (0, −1, 0, −1, 0, 1) ⊤ / √ 3, u 4 := (0, 0, −1, 0, −1, −1) ⊤ / √ 3.…”