We introduce a framework based on the Delsarte-Yudin linear programming approach for improving some universal lower bounds for the minimum energy of spherical codes of prescribed dimension and cardinality, and universal upper bounds on the maximal cardinality of spherical codes of prescribed dimension and minimum separation. Our results can be considered as next level universal bounds as they have the same general nature and imply, as the first level bounds do, necessary and sufficient conditions for their local and global optimality. We explain in detail our approach for deriving second level bounds. While there are numerous cases for which our method applies, we will emphasize the model examples of $24$ points ($24$-cell) and $120$ points ($600$-cell) on $\mathbb{S}^3$. In particular, we provide a new proof that the $600$-cell is universally optimal, and furthermore, we completely characterize the optimal linear programing polynomials of degree at most $17$ by finding two new polynomials, which together with the Cohn-Kumar's polynomial form the vertices of the convex hull that consists of all optimal polynomials. Our framework also provides a conceptual explanation of why polynomials of degree $17$ are needed to handle the $600$-cell via linear programming.

中文翻译：

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球形代码的界限：Levenshtein 框架解除

我们引入了一个基于 Delsarte-Yudin 线性规划方法的框架，用于改进规定维数和基数的球码的最小能量的一些通用下界，以及规定维数和最小间距的球码的最大基数的通用上限。我们的结果可以被视为下一级通用边界，因为它们具有相同的一般性质，并且与第一级边界一样，意味着它们的局部和全局最优性的充分必要条件。我们详细解释了我们导出二级边界的方法。虽然我们的方法适用于许多情况，但我们将强调 $\mathbb{S}^3$ 上的 $24$ 积分（$24$-cell）和 $120$ 积分（$600$-cell）的模型示例。特别是，我们提供了一个新的证明，证明 $600$-cell 是普遍最优的，此外，我们通过找到两个新的多项式，完整地刻画了阶数最多为 $17$ 的最优线性规划多项式，这两个多项式与 Cohn-Kumar 多项式一起形成了由所有最优多项式组成的凸包。我们的框架还提供了一个概念性解释，说明为什么需要 $17$ 次多项式来通过线性规划处理 $600$-cell。