Resolvable Triple Arrays

TL;DR

The paper introduces a new method for constructing triple arrays using 2-designs, producing the first examples of certain array types and proposing a new conjecture related to their existence.

Contribution

It presents a general construction technique for non-extremal triple arrays, introduces unordered triple arrays, and supports a new conjecture on their existence with enumerations and infinite families.

Findings

First examples of (21×15, 63)-triple arrays produced.

Enumeration of all resolvable (7×15, 35)-triple arrays.

All ((q+1)×q^2, q(q+1))-triple arrays are resolvable and linked to affine planes.

Abstract

We present a new construction of triple arrays by combining a symmetric 2-design with a resolution of another 2-design. This is the first general method capable of producing non-extremal triple arrays. We call the triple arrays which can be obtained in this way resolvable. We employ the construction to produce the first examples of $(21 \times 15, 63)$-triple arrays, and enumerate all resolvable $(7 \times 15, 35)$-triple arrays, of which there was previously only a single known example. An infinite subfamily of Paley triple arrays turns out to be resolvable. We also introduce a new intermediate object, unordered triple arrays, that are to triple arrays what symmetric 2-designs are to Youden rectangles, and propose a strengthening of Agrawal's long-standing conjecture on the existence of extremal triple arrays. For small parameters, we completely enumerate all unordered triple arrays, and use this data to corroborate the new conjecture. We construct several infinite families of resolvable unordered triple arrays, and, in particular, show that all $((q + 1) \times q^2, q(q + 1))$-triple arrays are resolvable and are in correspondence with finite affine planes of order $q$.