We characterize the cases of existence of spherical designs of an odd strength attaining the Fazekas–Levenshtein bound for covering and prove some of their properties. We also find all universal minima of the potential of regular spherical configurations in two new cases: the demihypercube on \(S^d\), \(d\ge 4\), and the \(2_{41}\) polytope on \(S^7\) (which is dual to the \(E_8\) lattice).

中文翻译：

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达到 Fazekas-Levenshtein 的奇数强度球形设计，覆盖范围和通用潜力最小值

我们描述了具有奇数强度的球形设计的存在情况，其达到了覆盖的 Fazekas-Levenshtein 边界，并证明了它们的一些特性。我们还发现了两种新情况下规则球形构型潜力的所有通用最小值：\(S^d\) 、 \ (d\ge 4\)上的半超立方体和\(2_{41}\)上的多胞形\(S^7\) （与\(E_8\)晶格对偶）。