2023/01/02 by Wang, Jiaxin, Fu, Fang-Wei, Wei, Yadi · 1 citation
#Combinatorics (math.CO) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT)
paper · doi:10.48550/arxiv.2301.00581
It is known that partial spreads is a class of bent partitions. In \citeAM2022Be,MP2021Be, two classes of bent partitions whose forms are similar to partial spreads were presented. In \citeAKM2022Ge, more bent partitions Γ1, Γ2, Γ1\bullet, Γ2\bullet, Θ1, Θ2 were presented from (pre)semifields, including the bent partitions given in \citeAM2022Be,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 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 C one-to-one correspond to vectorial dual-bent functions satisfying Condition A. We give an alternative proof that Γ1, Γ2, Γ1\bullet, Γ2\bullet, Θ1, Θ2 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: Vn(3)→ \mathbbF3 (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 \citeAM2022Be. 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 C in terms of partial difference sets.