
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 -triple arrays, and enumerate all resolvable -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,…
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