Small Youden Rectangles, Near Youden Rectangles, and Their Connections to Other Row-Column Designs

TL;DR

This paper systematically enumerates small Youden rectangles and near Youden rectangles, analyzes their symmetries and transformations, and explores their connections to other combinatorial designs like triple and sesqui arrays.

Contribution

It provides the first large-scale enumeration of small Youden rectangles and introduces near Youden rectangles, expanding understanding of their structure and relationships to other designs.

Findings

Enumerated all Youden rectangles for specific small parameters.

Identified near Youden rectangles with controlled column intersection properties.

Explored transformations linking these rectangles to other combinatorial designs.

Abstract

In this paper we first study $k \times n$ Youden rectangles of small orders. We have enumerated all Youden rectangles for a range of small parameter values, excluding the almost square cases where $k = n-1$, in a large scale computer search. In particular, we verify the previous counts for $(n,k) = (7,3), (7,4)$, and extend this to the cases $(11,5), (11,6), (13,4)$ and $(21,5)$. For small parameter values where no Youden rectangles exist, we also enumerate rectangles where the number of symbols common to two columns is always one of two possible values, differing by 1, which we call \emph{near Youden rectangles}. For all the designs we generate, we calculate the order of the autotopism group and investigate to which degree a certain transformation can yield other row-column designs, namely double arrays, triple arrays and sesqui arrays. Finally, we also investigate certain Latin rectangles with three possible pairwise intersection sizes for the columns and demonstrate that these can give rise to triple and sesqui arrays which cannot be obtained from Youden rectangles, using the transformation mentioned above.