Group testing via residuation and partial geometries

TL;DR

This paper introduces a novel approach to non-adaptive group testing using residuated pairs and partial geometries, providing efficient decoding schemes and constructing test matrices based on finite partial linear spaces, with applications to disease detection.

Contribution

It presents a new theoretical framework for group testing using residuation and partial geometries, along with practical constructions of test schemes based on incidence matrices.

Findings

Efficient decision scheme for infection pattern detection.

Construction of group testing schemes from incidence matrices.

Application to disease testing scenarios.

Abstract

The motivation for this paper comes from the ongoing SARS-CoV-2 Pandemic. Its goal is to present a previously neglected approach to non-adaptive group testing and describes it in terms of residuated pairs on partially ordered sets. Our investigation has the advantage, as it naturally yields an efficient decision scheme (decoder) for any given testing scheme. This decoder allows to detect a large amount of infection patterns. Apart from this, we devise a construction of good group testing schemes that are based on incidence matrices of finite partial linear spaces. The key idea is to exploit the structure of these matrices and make them available as test matrices for group testing. These matrices may generally be tailored for different estimated disease prevalence levels. As an example, we discuss the group testing schemes based on generalized quadrangles. In the context at hand, we state our results only for the error-free case so far. An extension to a noisy scenario is desirable and will be treated in a subsequent account on the topic.