Bent Partitions, Vectorial Dual-Bent Functions and Partial Difference Sets

Abstract

Bent partitions of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$V_{n}^{(p)}$ </tex-math></inline-formula> are quite powerful in constructing bent functions, vectorial bent functions and generalized bent functions, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$V_{n}^{(p)}$ </tex-math></inline-formula> is an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> -dimensional vector space over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {F}_{p}$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> is an even positive integer and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> is a prime. The classical examples of bent partitions are obtained from (partial) spreads. In Anbar and Meidl (2022) and Meidl and Pirsic (2021), two classes of bent partitions which are not obtained from (partial) spreads were presented. In Anbar et al. (2023), more bent partitions <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Gamma _{1}, \Gamma _{2}, \Gamma _{1}^{\bullet }, \Gamma _{2}^{\bullet }, \Theta _{1}, \Theta _{2}$ </tex-math></inline-formula> were presented from (pre)semifields, including the bent partitions given in Anbar and Meidl (2022) and Meidl and Pirsic (2021). In this paper, we investigate the relations between bent partitions and vectorial dual-bent functions. For any prime <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> , we show that one can generate certain bent partitions (called bent partitions satisfying Condition <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {C}$ </tex-math></inline-formula> ) from certain vectorial dual-bent functions (called vectorial dual-bent functions satisfying Condition A). In particular, when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> is an odd prime, we show that bent partitions satisfying Condition <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {C}$ </tex-math></inline-formula> one-to-one correspond to vectorial dual-bent functions satisfying Condition A. We give an alternative proof that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Gamma _{1}, \Gamma _{2}, \Gamma _{1}^{\bullet }, \Gamma _{2}^{\bullet }, \Theta _{1}, \Theta _{2}$ </tex-math></inline-formula> are bent partitions in terms of vectorial dual-bent functions. We present a secondary construction of vectorial dual-bent functions, which can be used to generate more bent partitions. We show that any weakly regular ternary bent function <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f: V_{n}^{(3)}\rightarrow \mathbb {F}_{3}$ </tex-math></inline-formula> ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> is even) of 2-form can generate a bent partition. When such <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f$ </tex-math></inline-formula> is weakly regular but not regular, the generated bent partition from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f$ </tex-math></inline-formula> is not coming from a normal bent partition, which answers an open problem proposed in Anbar and Meidl (2022). We give a sufficient condition on constructing partial difference sets from bent partitions, and when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> is an odd prime, we provide a characterization of bent partitions satisfying Condition <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {C}$ </tex-math></inline-formula> in terms of partial difference sets.

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