STOLARSKY'S INVARIANCE PRINCIPLE FOR FINITE METRIC SPACES
STOLARSKY'S INVARIANCE PRINCIPLE FOR FINITE METRIC SPACES
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DOI：
10.1112/mtk.12066
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发表时间：
2021-01-01
期刊：
影响因子：
0.8
通讯作者：
Barg, Alexander
中科院分区：
文献类型：
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作者：
Barg, Alexander
斯托拉斯基的不变性原理量化了度量空间与均匀分布的子集的偏差。它是针对球形组的经典得出的，最近在许多其他情况下进行了研究，揭示了各种形式的主要身份背后的一般结构。在这项工作中，我们考虑了有限的度量空间的情况，将子集的二次差异与距离分布的某些功能有关。我们的主要结果与锤子空间的不变性原理的具体形式有关。我们为代码的差异得出了表达式的几个等效版本，包括在krawtchouk基础上扩展差异和相关内核。在同一基数的所有子集中，具有最小的二次差异的代码自然可以被视为空间中的能量最小化子集。使用线性编程，我们在最小差异上找到了几个界限，并提供了最小化配置的示例。特别是，我们表明所有二进制完美代码都具有最小的差异。
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.