The existence spectrum of near triple arrays with four rows

Abstract

In the 1950s and 1960s, Agrawal introduced a class of experimental designs that later became known as triple arrays. Gordeev, Markström and Öhman proposed near triple arrays by relaxing all three intersection properties of triple arrays, allowing two values concentrated around the average intersection size, as well as two consecutive values for the replication numbers. They completely resolved the existence of near triple arrays with three rows, showing that there exists a $(3\times c,v)$-near triple array if and only if $v\geq c\geq 3$ except for $(c,v)\in\{(3,6),(4,6),(5,8)\}$. In this paper, we further investigate the existence of near triple arrays with four rows and prove that there exists a $(4\times c,v)$-near triple array if and only if $v\geq c\geq 4$ except for $(c,v)\in\{ (4,9),(5,7),(5,10),(6,8),(7,9),(10,12),(11,13)\}$.