论文标题
尖锐球形代码的离散潜力的通用最小值
Universal minima of discrete potentials for sharp spherical codes
论文作者
论文摘要
本文专门研究$ \ mathbb {r}^n $ in Sharp代码中的Sphere上的离散潜力。我们表明，大多数已知的尖锐代码的电势实现了先前由作者先前派生的球形$τ$ - 设计的通用下限，其中``通用''的意思是将其应用于包含内部产品的绝对单调功能的大量潜力。我们还将通用界限扩展到$ t $ - 设计以及由球形配置消失的矩确定的相关多项式子空间，从而获得了iCosahedron，dodecahedron的最小值，以及来自$ e_8 $ and e_8 $和leech晶格的尖锐代码。为此，我们研究了Gegenbauer polyenmials的某些子空间$ p^{（n）} _ j $的正交公式，我们称之为PULB子空间，尤其是那些具有基础$ \ {p_j^{（n）} {（n）} \} _ = 0} _ = 0}^{2k+2k+2k+2k+2} {2k+2k+2k+2k+2} {（n）} p_j^{（n） \ {p_ {2k}^{（n）} \}。$此外，对于具有$ h^{（τ+1）} <0 $的电势，我们证明了强尖锐的代码和强尖锐的敏锐代码和对抗的敏锐代码达到了通用界限，并且它们的最小值出现在代码点上。当潜在$ h $满足$ h^{（i）} \ geq 0 $，$ i = 1，\ dots，15 $和$ h^{（16）} \ leq 0时
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 $τ$-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, 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 $P^{(n)}_j$ which we call PULB subspaces, particularly those having basis $\{P_j^{(n)}\}_{j=0}^{2k+2}\setminus \{P_{2k}^{(n)}\}.$ Furthermore, for potentials with $h^{(τ+1)}<0$ 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 $h^{(i)}\geq 0$, $i=1,\dots,15$, and $h^{(16)}\leq 0.$
