Stolarsky's invariance principle for finite metric spaces

TL;DR

This paper extends Stolarsky's invariance principle to finite metric spaces, especially Hamming space, relating subset discrepancy to distance distributions, and identifies minimal discrepancy codes including perfect codes.

Contribution

It introduces a concrete form of the invariance principle for finite metric spaces, derives discrepancy formulas using Krawtchouk polynomials, and applies linear programming to find minimal discrepancy codes.

Findings

Binary perfect codes have minimal quadratic discrepancy.

Derived bounds on minimal discrepancy using linear programming.

Established equivalences of discrepancy expressions in Hamming space.

Abstract

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.