The concept of formal duality was proposed by Cohn, Kumar and Schurmann, which reflects a remarkable symmetry among energy-minimizing periodic configurations. This formal duality was later on translated into a purely combinatorial property by Cohn, Kumar, Reiher and Schurmann, where the corresponding combinatorial objects were called formally dual pairs. Almost all known examples of primitive formally dual pairs satisfy that the two subsets have the same size. Indeed, prior to this work, there was only one known example having subsets with unequal sizes in $\mathbb{Z}_2 \times \mathbb{Z}_4^2$. Motivated by this example, we propose a lifting construction framework and a recursive construction framework, which generate new primitive formally dual pairs from known ones. As an application, for $m \ge 2$, we obtain $m+1$ pairwise inequivalent primitive formally dual pairs in $\mathbb{Z}_2 \times \mathbb{Z}_4^{2m}$, which have subsets with unequal sizes.

中文翻译：

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具有不等大小子集的原始形式对偶对的构造

形式对偶的概念是由 Cohn、Kumar 和 Schurmann 提出的，它反映了能量最小化周期配置之间显着的对称性。这种形式对偶后来被 Cohn、Kumar、Reiher 和 Schurmann 翻译成纯粹的组合性质，其中相应的组合对象被称为形式对偶对。几乎所有已知的原始形式对偶对的例子都满足两个子集具有相同的大小。事实上，在这项工作之前，只有一个已知的例子在 $\mathbb{Z}_2 \times \mathbb{Z}_4^2$ 中有大小不等的子集。受这个例子的启发，我们提出了一个提升构造框架和一个递归构造框架，它们从已知的生成新的原始形式对偶对。作为一个应用程序，对于 $m \ge 2$，