Abstract
We propose a hierarchy of k-point bounds extending the Delsarte–Goethals–Seidel linear programming 2-point bound and the Bachoc–Vallentin semidefinite programming 3-point bound for spherical codes. An optimized implementation of this hierarchy allows us to compute 4, 5, and 6-point bounds for the maximum number of equiangular lines in Euclidean space with a fixed common angle.
Highlights
Given D ⊆ [−1, 1), a subset C of the unit sphere Sn−1 = { x ∈ Rn : x = 1 } is a spherical D-code if x · y ∈ D for all distinct x, y ∈ C, where x · y is the Euclidean inner product between x and y
A fundamental tool for computing upper bounds for A(n, D) is the linear programming bound of Delsarte et al [13], which is an adaptation of the Delsarte bound [12] to the sphere
Bachoc and Vallentin [2] extended the linear programming bound to a 3-point bound by taking into account interactions between triples of points, extending the three-point bound by Schrijver [44] for binary codes
Summary
The linear programming bound was one of the first nontrivial upper bounds for the kissing number and is the optimal value of a convex optimization problem It is a 2-point bound, because it takes into account interactions between pairs of points on the sphere: pairs {x, y} with x · y ∈/ D correspond to constraints in the optimization problem. For the case where D is finite there is no such obstruction, and though our hierarchy is not as strong, in theory, as the Lasserre hierarchy, it is computationally less expensive This allows us to use it to compute 4, 5, and 6-point bounds for the maximum number of equiangular lines with a certain angle, a problem that corresponds to the case |D| = 2. This implementation could be of interest to others working on similar problems
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