Universal minima of discrete potentials for sharp spherical codes

Abstract

This article is devoted to the study of discrete potentials on the sphere in ${\mathbb{R}}^n$ for sharp codes. We show that the potentials of most of the known sharp codes attain the universal lower bounds for polarization for spherical $\tau$-designs previously derived by the authors, where “universal” is meant in the sense of applying to a large class of potentials that includes absolutely monotone functions of inner products. We also extend our universal bounds to $T$-designs and the associated polynomial subspaces determined by the vanishing moments of spherical configurations, and thus obtain the minima for the icosahedron, the dodecahedron, and sharp codes coming from $E_8$ and the Leech lattice. For this purpose, we investigate quadrature formulas for certain subspaces of Gegenbauer polynomials $P_j^{(n)}$ which we call PULB subspaces, particularly those having basis $\{P_j^{(n)}{\}}_{j=0}^{2k+2}\setminus \{P_{2k}^{(n)}\}.$ Furthermore, for potentials with $h^{(\tau +1)}<0$, we prove that the strong sharp codes and the antipodal sharp codes attain the universal bounds and their minima occur at points of the codes. The same phenomenon is established for the $600$-cell when the potential $h$ satisfies $h^{(i)}\ge 0$, $i=1,\dots ,15$, and $h^{(16)}\le 0$.