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 $q$.

翻译：我们提出了一种通过结合一个对称2-设计与另一个2-设计的解来构造三重阵列的新方法。这是首个能够生成非极端三重阵列的通用方法。我们将通过这种方式获得的三重阵列称为可解的。我们利用该构造生成了首个$(21 \times 15, 63)$-三重阵列的实例，并枚举了所有可解的$(7 \times 15, 35)$-三重阵列——此前该参数仅有一个已知实例。Paley三重阵列的一个无限子族被证明是可解的。我们还引入了一个新的中间对象——无序三重阵列，它对于三重阵列而言，正如对称2-设计对于尤登矩形那样，并提出了对阿格拉沃尔关于极端三重阵列存在性的长期猜想的一个加强版本。对于小参数，我们完全枚举了所有无序三重阵列，并利用这些数据验证了新猜想。我们构造了若干无限族的可解无序三重阵列，特别地，证明了所有$((q + 1) \times q^2, q(q + 1))$-三重阵列都是可解的，且与$q$阶有限仿射平面一一对应。