Constructions of primitive formally dual pairs having subsets with unequal sizes. Issue 12 (29th September 2019)

• Record Type: Journal Article
• Authors: Li, Shuxing; Pott, Alexander
• Is Part Of: Journal of combinatorial designs. Volume 27:Issue 12(2019:Dec.)
• Page Start: 703
• Page End: 733
• Publication Date: 2019-09-29
• Subjects: character sum -- energy minimization -- formal duality -- inequivalence -- lifting construction -- periodic configuration -- primitive formally dual pair -- recursive construction; Combinatorial designs and configurations -- Periodicals; Configurations et schémas combinatoires -- Périodiques; 511.6
• DOI: 10.1002/jcd.21674
• Languages: English
• ISSNs: 1063-8539

Abstract

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. Motivated by this surprising application on the energy minimization problem, we focus on the algebraic constructions of primitive formally dual pairs. It is worth noting that 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 derived from computer search, which had subsets with unequal sizes in Z 2 × Z 4 2 . Inspired 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 ≥ 2, we obtain m + 1 pairwise inequivalent primitive formally dual pairs in Z 2 × Z 4 2 m, which have subsets with unequal sizes.