Bent Partitions, Vectorial Dual-Bent Functions and Partial Difference Sets
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It is known that partial spreads is a class of bent partitions. In <cit.>, two classes of bent partitions whose forms are similar to partial spreads were presented. In <cit.>, more bent partitions Γ_1, Γ_2, Γ_1^∙, Γ_2^∙, Θ_1, Θ_2 were presented from (pre)semifields, including the bent partitions given in <cit.>. 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 Γ_1, Γ_2, Γ_1^∙, Γ_2^∙, Θ_1, Θ_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)→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 <cit.>. 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.
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