向量广义Maiorana-McFarland弯曲函数与非弱正则分量构成的结合方案
Association schemes from vectorial generalized Maiorana-McFarland bent functions with non-weakly regular components
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中文总结 AI 辅助
本文研究向量广义Maiorana-McFarland弯曲函数，证明其水平集划分的Fourier-自反性，从而首次从非弱正则分量向量弯曲函数构造出多类平移结合方案，并给出融合构造与反例。
中文摘要 AI 辅助
设$p$为奇素数，$q=p^k$，$r=p^s$且$s\mid k$。我们研究定义在$V_n\times\mathbb{F}_q\times\mathbb{F}_q$上的$\mathbb{F}_r$-值广义Maiorana-McFarland函数$H(x,y,z)=P^{(z)}(x)+\operatorname{Tr}_s^k(yz)$。其中$P^{(z)}$为具有弱正则分量的向量对偶弯曲函数，在$\mathbb{F}_q^*$中$\mathbb{F}_r^*$的陪集上取常值，且当$z\ne 0$时，关于映射$t\mapsto t^{-1}$为反演型。在关于$P^{(0)}$的温和条件下，我们证明$H$的$\mathbb{F}_r$-值水平集按相关坐标的$\mathbb{F}_r$-线细化后的划分是Fourier-自反的。因此它诱导一个具有$\frac{q-1}{r-1}(r+1)+r$个类的平移结合方案（在退化情形下少一个类），当分量为偶函数时该方案是对称的。当$s=1$时，这恢复了最近的标量构造；当$s=k$时，它从分量可能全部非弱正则的向量弯曲函数产生$(2q+1)$类方案。我们还沿$\mathbb{F}_r$-子空间构造融合，得到$(2r+1)$类和$(3r+2)$类方案以及不同水平之间的连接。反例表明反演、陪集常值、子空间和弱正则性假设一般不能被省略。据我们所知，这是首次从具有非弱正则分量的向量弯曲函数获得结合方案。
英文摘要
Let $p$ be an odd prime, $q=p^k$, and $r=p^s$ with $s\mid k$. We study $\mathbb{F}_r$-valued generalized Maiorana-McFarland functions $H(x,y,z)=P^{(z)}(x)+\operatorname{Tr}_s^k(yz)$ on $V_n\times\mathbb{F}_q\times\mathbb{F}_q$. The ingredients $P^{(z)}$ are vectorial dual-bent functions with weakly regular components, constant on the cosets of $\mathbb{F}_r^*$ in $\mathbb{F}_q^*$, and, for $z\ne 0$, of inversion type with respect to $t\mapsto t^{-1}$. Under a mild condition on $P^{(0)}$, we prove that the partition into the $\mathbb{F}_r$-valued level sets of $H$, refined according to the $\mathbb{F}_r$-lines of the relevant coordinates, is Fourier-reflexive. It therefore induces a translation association scheme with $\frac{q-1}{r-1}(r+1)+r$ classes (one fewer in a degenerate case), symmetric when the ingredients are even. For $s=1$ this recovers recent scalar constructions, while for $s=k$ it yields $(2q+1)$-class schemes from vectorial bent functions whose components may all be non-weakly regular. We also construct fusions along $\mathbb{F}_r$-subspaces, obtaining $(2r+1)$- and $(3r+2)$-class schemes and connections between different levels. Counterexamples show that the inversion, coset-constancy, subspace, and weak-regularity hypotheses cannot in general be omitted. To our knowledge, these are the first association schemes obtained from vectorial bent functions with non-weakly regular components.
发表机构
- University of South Florida（南佛罗里达大学）
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