球码势能的泛下界
Universal lower bounds for potential energy of spherical codes
摘要：
我们推导并研究了有限球面点集（球面码）势能的下界。我们的界限是最佳的，在以下意义上-他们不能通过采用多项式的相同或更低的次数在Delsarte-Yudin方法。然而，改进有时是可能的，我们提供了一个充要条件的存在，这样的更好的界限。我们所有的边界都可以用一种不依赖于势函数的统一方式得到，只要势是由点对之间的内积的绝对单调函数给出的，这就是我们称它们为泛函数的原因。我们还建立了维数为$n$和基数为$N$的码不是LP-泛最优的准则，例如我们证明了Ballinger等人证明的两个泛最优码不是LP-泛最优的.
英文摘要：
We derive and investigate lower bounds for the potential energy of finite spherical point sets (spherical codes). Our bounds are optimal in the following sense -- they cannot be improved by employing polynomials of the same or lower degrees in the Delsarte-Yudin method. However, improvements are sometimes possible and we provide a necessary and sufficient condition for the existence of such better bounds. All our bounds can be obtained in a unified manner that does not depend on the potential function, provided the potential is given by an absolutely monotone function of the inner product between pairs of points, and this is the reason for us to call them universal. We also establish a criterion for a given code of dimension $n$ and cardinality $N$ not to be LP-universally optimal, e.g. we show that two codes conjectured by Ballinger et al to be universally optimal are not LP-universally optimal.
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