Triple arrays from difference sets

TL;DR

This paper investigates constructing triple arrays from difference sets, establishing conditions for abelian and non-abelian cases, and introduces a new infinite family of triple arrays with computational exploration.

Contribution

It provides a new characterization of when difference sets yield triple arrays, including an infinite family and conditions for non-abelian cases, with a direct construction method.

Findings

Abelian difference sets with -1 multiplier produce triple arrays

A new infinite family of triple arrays is constructed

No non-abelian difference sets satisfying the conditions were found in computer searches

Abstract

This paper addresses the question whether triple arrays can be constructed from Youden squares developed from difference sets. We prove that if the difference set is abelian, then having $-1$ as multiplier is both a necessary and sufficient condition for the construction to work. Using this, we are able to give a new infinite family of triple arrays. We also give an alternative and more direct version of the construction, leaving out the intermediate step via Youden squares. This is used when we analyse the case of non-abelian difference sets, for which we prove a sufficient condition for giving triple arrays. We do a computer search for such non-abelian difference sets, but have not found any examples satisfying the given condition.