Finding and investigating exact spherical codes

In this paper we present the results of computer searches using a variation of an energy minimization algorithm used by Kottwitz for finding good spherical codes. We prove that exact codes exist by representing the inner products between the vectors as algebraic numbers. For selected interesting cases, we include detailed discussion of the configurations. Of particular interest are the 20-point code in $\mathbb{R}^6$ and the 24-point code in $\mathbb{R}^7$, which are both the union of two cross polytopes in parallel hyperplanes. Finally, we catalogue all of the codes we have found.