Plateaued functions on finite abelian groups and partial geometric difference sets

As a generalization of plateaued functions on finite fields and bent functions (perfect nonlinear functions) on finite abelian groups, plateaued functions on finite abelian groups were introduced in [B. Xu, Plateaued functions, partial geometric difference sets, and partial geometric designs, J. Com...

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Vydáno v:Journal of combinatorial designs Ročník 30; číslo 4; s. 220 - 250
Hlavní autor: Xu, Bangteng
Médium: Journal Article
Jazyk:angličtina
Vydáno: Hoboken Wiley Subscription Services, Inc 01.04.2022
Témata:
Combinatorial analysis
cyclotomic fields
derivatives
Fields (mathematics)
Group theory
Mathematical analysis
partial geometric designs
partial geometric difference sets
Plateaued functions
Set theory
tensor products
Tensors
ISSN:1063-8539, 1520-6610
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Shrnutí:As a generalization of plateaued functions on finite fields and bent functions (perfect nonlinear functions) on finite abelian groups, plateaued functions on finite abelian groups were introduced in [B. Xu, Plateaued functions, partial geometric difference sets, and partial geometric designs, J. Combin. Des. 27 (2019), 756–783]. In this paper, we continue the research in the paper mentioned above. We will obtain various characterizations of plateaued functions; these characterizations establish close connections between plateaued functions and some combinatorial objects: partial geometric difference sets and related partial geometric difference families. Then we introduce the complementary matrix and Cayley matrix for a subset of a finite group and use them to characterize partial geometric difference sets. As applications, we will show how to construct directed strongly regular graphs from partial geometric difference sets and establish a natural relation between partial geometric difference sets and partial geometric designs. The tensor product of a group algebra and a cyclotomic field is an important tool for our discussions.
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ISSN:1063-8539
1520-6610
DOI:10.1002/jcd.21821