消失平面：函数平面性的组合观点及其应用
Vanishing Flats: A Combinatorial Viewpoint on the Planarity of Functions and Their Application
摘要：
对于从$\mathbb{F}_2^n$到$\mathbb{F}_2^n$的函数$f$，$f$的平面性通常通过其微分均匀性和微分谱来测量。本文提出了消失面的概念，为平面性问题提供了一种组合的观点。首先，$f$的消失平面的数目可以被视为$f$与几乎完美非线性函数集合之间的距离的度量。在某些情况下，消失平面的数量充当微分均匀性和微分光谱之间的“中间”概念，其包含比微分均匀性更多的信息，但是比微分光谱更少。第二，消失平面的集合形成称为部分四元组系统的组合配置，因为它传达了关于$f$的详细结构信息。我们开始这项研究，考虑消失的单位和部分四元系统与单项式和Dembowski-Ostrom多项式的数量。此外，我们提出了一个应用程序的消失单位分割成不相交的等维仿射空间的向量空间。最后，我们提出了一些进一步的问题和挑战。
英文摘要：
For a function $f$ from $\mathbb{F}_2^n$ to $\mathbb{F}_2^n$, the planarity of $f$ is usually measured by its differential uniformity and differential spectrum. In this paper, we propose the concept of vanishing flats, which supplies a combinatorial viewpoint on the planarity. First, the number of vanishing flats of $f$ can be regarded as a measure of the distance between $f$ and the set of almost perfect nonlinear functions. In some cases, the number of vanishing flats serves as an "intermediate" concept between differential uniformity and differential spectrum, which contains more information than differential uniformity, however less than the differential spectrum. Secondly, the set of vanishing flats forms a combinatorial configuration called partial quadruple system, since it convey detailed structural information about $f$. We initiate this study by considering the number of vanishing flats and the partial quadruple systems associated with monomials and Dembowski-Ostrom polynomials. In addition, we present an application of vanishing flats to the partition of a vector space into disjoint equidimensional affine spaces. We conclude the paper with several further questions and challenges.
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