Stolarsky's invariance principle quantifies the deviation of a subset of a metric space from the uniform distribution. Classically derived for spherical sets, it has been recently studied in a number of other situations, revealing a general structure behind various forms of the main identity. In this work we consider the case of finite metric spaces, relating the quadratic discrepancy of a subset to a certain function of the distribution of distances in it. Our main results are related to a concrete form of the invariance principle for the Hamming space. We derive several equivalent versions of the expression for the discrepancy of a code, including expansions of the discrepancy and associated kernels in the Krawtchouk basis. Codes that have the smallest possible quadratic discrepancy among all subsets of the same cardinality can be naturally viewed as energy minimizing subsets in the space. Using linear programming, we find several bounds on the minimal discrepancy and give examples of minimizing configurations. In particular, we show that all binary perfect codes have the smallest possible discrepancy.

中文翻译：

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有限度量空间的 STOLARSKY 不变性原理

Stolarsky 的不变性原理量化了度量空间的子集与均匀分布的偏差。它是球形集的经典推导，最近在许多其他情况下进行了研究，揭示了主要身份的各种形式背后的一般结构。在这项工作中，我们考虑有限度量空间的情况，将子集的二次差异与其中距离分布的某个函数相关联。我们的主要结果与汉明空间不变性原理的具体形式有关。我们导出了代码差异表达式的几个等效版本，包括差异的扩展和 Krawtchouk 基础中的相关内核。在具有相同基数的所有子集之间具有最小可能的二次差异的代码可以自然地被视为空间中的能量最小化子集。使用线性规划，我们找到了最小差异的几个界限，并给出了最小化配置的例子。特别是，我们表明所有二进制完美代码都具有最小的可能差异。