Uniform exclude distributions of Sidon sets

Abstract

A Sidon set $S$ in $\mathbb{F}_2^n$ is a set such that the pairwise sums of distinct points are all distinct. The exclude points of a Sidon set $S$ are the sums of three distinct points in $S$, and the exclude multiplicity of a point in $\mathbb{F}_2^n \setminus S$ is the number of such triples in $S$ it is equal to. We call the function $d_S \colon \mathbb{F}_2^n \setminus S \to \mathbb{Z}_{\geq 0}$ taking points in $\mathbb{F}_2^n \setminus S$ to their exclude multiplicity the exclude distribution of $S$. We say that $d_S$ is uniform on $\mathcal{P}$ if $\mathcal{P}$ is an equally-sized partition $\mathcal{P}$ of $\mathbb{F}_2^n \setminus S$ such that $d_S$ takes the same values an equal number of times on every element of $\mathcal{P}$. In this paper, we use APN plateaued functions with all component functions unbalanced to construct Sidon sets $S$ in $(\mathbb{F}_2^n)^2$ whose exclude distributions are uniform on natural partitions of $(\mathbb{F}_2^n)^2 \setminus S$ into $2^n$ elements. We use this result and a result of Carlet to determine exactly what values the exclude distributions of the graphs of the Gold and Kasami functions take and how often they take these values.