Large Equiangular Sets of Lines in Euclidean Space

Abstract

A construction is given of ${{2}\over {9}} (d+1)^2$ equiangular lines in Euclidean $d$-space, when $d = 3 \cdot 2^{2t-1}-1$ with $t$ any positive integer. This compares with the well known "absolute" upper bound of ${{1}\over {2}} d(d+1)$ lines in any equiangular set; it is the first known constructive lower bound of order $d^2$ .

Equiangular lines and spherical codes in Euclidean space

“…We can assume α ′ k ≥ β 2 /2, otherwise choosing c β,k > (k + 1) (k+1)t (4β −2 + 4) we are done by the first case considered above. Since α ′ r = (α −1 r + t) −1 < β ′ , the computation in (6) implies that there exists ℓ > 1 such that α ℓ−1 < α 2 ℓ /2. Choosing c β,k > (k + 1) (k+1)t 2kc β,ℓ−1 c β ′ ,k−ℓ+1 we are done by the second case considered above.…”

Equiangular line systems and switching classes containing regular graphs

“…Let d 1, let S be a Seidel matrix of order n 2 with smallest eigenvalue λ 0 of multiplicity n − d 1, and let µ be the closest even integer to −λ 0 (n − d)/d. Suppose λ 2 0 > d, n = ⌊d(λ 2 0 − 1)/(λ 2 0 − d)⌋, and that (7) (n − d)(λ 0 − µ) 2 − nµ 2 + d > n(n − 1). Proof.…”

Linked systems of symmetric designs

“…This construction is a generalization of de Caen's construction [4] using the Cameron-Seidel scheme. Using the above idempotents, we wish to make a positive semi-definite matrix with low rank with constant diagonal and only one possible magnitude of the main diagonal.…”