A direct construction of primitive formally dual pairs having subsets with unequal sizes

Abstract

The concept of formal duality was proposed by Cohn, Kumar and Schürmann, which reflects a remarkable symmetry among energy-minimizing periodic configurations. This formal duality was later translated into a purely combinatorial property by Cohn, Kumar, Reiher and Schürmann, where the corresponding combinatorial objects were called formally dual pairs. So far, except the results presented in Li and Pott (J. Combin. Des., in press), we have little information about primitive formally dual pairs having subsets with unequal sizes. In this paper, we propose a direct construction of primitive formally dual pairs having subsets with unequal sizes in $\mathbb {Z}_{2} \times \mathbb {Z}_{4}^{2m}$, where m ≥ 1. This construction recovers an infinite family obtained in Li and Pott (J. Combin. Des., in press), which was derived by employing a recursive approach. Although the resulting infinite family was known before, the idea of the direct construction is new and provides more insights which were not known from the recursive approach.