Designs in finite metric spaces: a probabilistic approach

TL;DR

This paper introduces a probabilistic framework for designs in finite metric spaces, generalizing existing concepts, and derives limit laws for weight distributions in combinatorial structures as their strength increases.

Contribution

It defines a new notion of designs in finite metric spaces, extending the concept from distance-regular graphs, and applies approximation theory to analyze their distributional properties.

Findings

Derived limit laws for weight distributions of binary orthogonal arrays.

Established analogous results for combinatorial designs of increasing strength.

Provided an approximation method for their cumulative distribution functions.

Abstract

A finite metric space is called here distance degree regular if its distance degree sequence is the same for every vertex. A notion of designs in such spaces is introduced that generalizes that of designs in $Q$-polynomial distance-regular graphs. An approximation of their cumulative distribution function, based on the notion of Christoffel function in approximation theory is given. As an application we derive limit laws on the weight distributions of binary orthogonal arrays of strength going to infinity. An analogous result for combinatorial designs of strength going to infinity is given.