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A direct construction of primitive formally dual pairs having subsets with unequal sizes. (English) Zbl 1443.05032

Summary: The concept of formal duality was proposed by H. Cohn et al. [“Ground states and formal duality relations in the Gaussian core model”, Phys. Rev. E 80, No. 6, Article ID 061116, 7 p. (2009; doi:10.1103/PhysRevE.80.061116)], which reflects a remarkable symmetry among energy-minimizing periodic configurations. This formal duality was later translated into a purely combinatorial property by H. Cohn et al. [Contemp. Math. 625, 123–140 (2014; Zbl 1331.05056)], where the corresponding combinatorial objects were called formally dual pairs. So far, except the results presented in [the authors, “Constructions of primitive formally dual pairs having subsets with unequal sizes”, J. Combin. Des. (to appear)], 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 \(m \geq 1\). This construction recovers an infinite family obtained in [the authors, loc. cit.], 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.

MSC:

05B40 Combinatorial aspects of packing and covering
52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
20K01 Finite abelian groups

Citations:

Zbl 1331.05056
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References:

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