Bounds for spherical codes: The Levenshtein framework lifted

Based on the Delsarte-Yudin linear programming approach, we extend Levenshtein’s framework to obtain lower bounds for minimum $h$-energy of spherical codes prescribed dimension and cardinality, upper maximal cardinality separation. These are universal in sense that they hold a large class potentials $h$ Levenshtein. Moreover, attaining universally optimal Cohn-Kumar. Referring Levenshtein energy authors as “first level”, our results can be considered “next level” have same general nature imply necessary sufficient conditions their local global optimality. For this purpose, introduce notion Universal Lower Bound space (ULB-space), satisfies certain quadrature interpolation properties. While there numerous cases which method applies, will emphasize model examples $24$ points ($24$-cell) $120$ ($600$-cell) $\mathbb {S}^3$. In particular, provide new proof $600$-cell is optimal, so doing, derive optimality larger than absolutely monotone by