Vanishing Flats: A Combinatorial Viewpoint on the Planarity of Functions and Their Application

TL;DR

This paper introduces the concept of vanishing flats as a combinatorial tool to analyze the planarity of functions over finite fields, providing a new perspective that bridges differential uniformity and spectrum.

Contribution

It proposes vanishing flats as a novel combinatorial measure, linking function structure to geometric configurations and applications in vector space partitioning.

Findings

Vanishing flats quantify the distance to almost perfect nonlinear functions.

Partial quadruple systems encode structural information about functions.

Application to partitioning vector spaces into affine subspaces.

Abstract

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.