Enumeration of Row-Column Designs

TL;DR

This paper fully enumerates various row-column designs, introduces new design types, and explores their properties and connections, providing both computational and theoretical insights into their structure and existence.

Contribution

It introduces two new design types, mono arrays and AO-arrays, and provides complete enumeration, existence proofs, and explicit constructions for several classes of row-column designs.

Findings

Complete enumeration of certain row-column designs up to isotopism.

Introduction and characterization of mono arrays and AO-arrays.

Non-existence results for some parameter sets and explicit constructions for others.

Abstract

We computationally completely enumerate a number of types of row-column designs up to isotopism, including double, sesqui and triple arrays as known from the literature, and two newly introduced types that we call mono arrays and AO-arrays. We calculate autotopism group sizes for the designs we generate. For larger parameter values, where complete enumeration is not feasible, we generate examples of some of the designs, and generate exhaustive lists of admissible parameters. For some admissible parameter sets, we prove non-existence results. We also give some explicit constructions of sesqui arrays, mono arrays and AO-arrays, and investigate connections to Youden rectangles and binary pseud Youden designs.