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- W4313484313 abstract "It is known that partial spreads is a class of bent partitions. In cite{AM2022Be,MP2021Be}, two classes of bent partitions whose forms are similar to partial spreads were presented. In cite{AKM2022Ge}, more bent partitions $Gamma_{1}, Gamma_{2}, Gamma_{1}^{bullet}, Gamma_{2}^{bullet}, Theta_{1}, Theta_{2}$ were presented from (pre)semifields, including the bent partitions given in cite{AM2022Be,MP2021Be}. In this paper, we investigate the relations between bent partitions and vectorial dual-bent functions. For any prime $p$, we show that one can generate certain bent partitions (called bent partitions satisfying Condition $mathcal{C}$) from certain vectorial dual-bent functions (called vectorial dual-bent functions satisfying Condition A). In particular, when $p$ is an odd prime, we show that bent partitions satisfying Condition $mathcal{C}$ one-to-one correspond to vectorial dual-bent functions satisfying Condition A. We give an alternative proof that $Gamma_{1}, Gamma_{2}, Gamma_{1}^{bullet}, Gamma_{2}^{bullet}, Theta_{1}, Theta_{2}$ are bent partitions. We present a secondary construction of vectorial dual-bent functions, which can be used to generate more bent partitions. We show that any ternary weakly regular bent function $f: V_{n}^{(3)}rightarrow mathbb{F}_{3}$ ($n$ even) of $2$-form can generate a bent partition. When such $f$ is weakly regular but not regular, the generated bent partition by $f$ is not coming from a normal bent partition, which answers an open problem proposed in cite{AM2022Be}. We give a sufficient condition on constructing partial difference sets from bent partitions, and when $p$ is an odd prime, we provide a characterization of bent partitions satisfying Condition $mathcal{C}$ in terms of partial difference sets." @default.
- W4313484313 created "2023-01-06" @default.
- W4313484313 creator A5036949743 @default.
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- W4313484313 date "2023-01-02" @default.
- W4313484313 modified "2023-09-30" @default.
- W4313484313 title "Bent Partitions, Vectorial Dual-Bent Functions and Partial Difference Sets" @default.
- W4313484313 doi "https://doi.org/10.48550/arxiv.2301.00581" @default.
- W4313484313 hasPublicationYear "2023" @default.
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