Links between orthogonal arrays, association schemes and partial geometric designs

TL;DR

This paper explores the deep connections between orthogonal arrays, association schemes, and partial geometric designs, revealing new characterizations and infinite families of directed strongly regular graphs.

Contribution

It establishes novel links between these combinatorial structures and provides characterizations and infinite families of directed strongly regular graphs.

Findings

Partial geometric designs can be derived from certain association schemes and orthogonal arrays.

Characterizations of graphs, schemes, and arrays in terms of partial geometric designs are provided.

Infinite families of directed strongly regular graphs are constructed from the designs.

Abstract

In this paper, we show how certain three-class association schemes and orthogonal arrays give rise to partial geometric designs. We also investigate the connections between partial geometric designs and certain regular graphs having three or four distinct eigenvalues, three-class association schemes, orthogonal arrays of strength two and particular linear codes. We give various characterizations of these graphs, association schemes and orthogonal arrays in terms of partial geometric designs. We also give a list of infinite families of directed strongly regular graphs arising from the partial geometric designs obtained in this paper.