In this paper, we study two special subsets of a finite field of odd characteristics associated with non-weakly regular bent functions. We show that those subsets associated to non-weakly regular even bent functions in the GMMF class (see Çesmelioğlu et al. Finite Fields Appl. 24, 105–117 2013) are never partial difference sets (PDSs), and are PDSs if and only if they are trivial subsets. Moreover, we analyze the two known sporadic examples of non-weakly regular ternary bent functions given in Helleseth and Kholosha (IEEE Trans. Inf. Theory 52(5), 2018–2032 2006, Cryptogr. Commun. 3(4), 281–291 2011). We observe that corresponding subsets are non-trivial partial difference sets. We show that they are the union of some cyclotomic cosets and so correspond to 2-class fusion schemes of a cyclotomic scheme. We also present a further construction giving non-trivial PDSs from certain p-ary functions which are not bent functions.

中文翻译：

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由非弱正则弯曲函数产生的强正则图

在本文中，我们研究了与非弱正则弯曲函数相关的奇特特性有限域的两个特殊子集。我们发现，在相关联的非正规弱甚至弯曲功能的子集GMMF类（见Çesmelioğlu等有限域申请24，105-117 2013）从不偏差集（电气传动系统），并且是电气传动系统的当且仅如果它们是琐碎的子集。此外，我们的分析中Helleseth和Kholosha（硕士论文。INF组成。理论给定的非弱正三元弯曲函数的两种已知的零星实例52（5），2018至32年2006年，Cryptogr。COMMUN。3（4），281- 291 2011）。我们观察到相应的子集是不平凡的偏差集。我们证明它们是某些环原子集的联合，因此对应于一个环原子方案的2类融合方案。我们还提出了进一步的构造，该构造从某些pary函数（不是弯曲函数）提供了非平凡的PDS。