Semidefinite Programming Bounds for Spherical Three-Distance Sets

A spherical three-distance set is a finite collection $X$ of unit vectors in $\mathbb{R}^{n}$ such that the set of the distances between any two distinct vectors has cardinality three. We use the semidefinite programming method to improve the upper bounds for the cardinalities of spherical three-distance sets in the Euclidean spaces of several dimensions. We obtain better bounds in $\mathbb{R}^7$, $\mathbb{R}^{20}$, $\mathbb{R}^{21}$, $\mathbb{R}^{23}$, $\mathbb{R}^{24}$ and $\mathbb{R}^{25}$. In particular, we prove that the maximum cardinality of a spherical three-distance set in $\mathbb R^{23}$ is $2300$.