Finding and Investigating Exact Spherical Codes

Abstract

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 $\mathbb{R}^7$, each of which is the union of two cross-polytopes in parallel hyperplanes. Finally, we catalogue all of the codes we have found.