A Sidon set $M$ is a subset of $\mathbb{F}_2^t$ such that the sum of four distinct elements of $M$ is never 0. The goal is to find Sidon sets of large size. In this note we show that the graphs of almost perfect nonlinear (APN) functions with high linearity can be used to construct large Sidon sets. Thanks to recently constructed APN functions $\mathbb{F}_2^8\to \mathbb{F}_2^8$ with high linearity, we can construct Sidon sets of size 192 in $\mathbb{F}_2^{15}$, where the largest sets so far had size 152. Using the inverse and the Dobbertin function also gives larger Sidon sets as previously known. Each of the new large Sidon sets $M$ in $\mathbb{F}_2^t$ yields a binary linear code with $t$ check bits, minimum distance 5, and a length not known so far. Moreover, we improve the upper bound for the linearity of arbitrary APN functions.

翻译：西顿集 $M$ 是 $\mathbb{F}_2^t$ 的一个子集，其任意四个不同元素之和永不为 0。目标是寻找大尺寸的西顿集。本文指出，具有高线性度的几乎完美非线性（APN）函数的图可用于构造大西顿集。得益于近期构造出的具有高线性度的 APN 函数 $\mathbb{F}_2^8\to \mathbb{F}_2^8$，我们能够在 $\mathbb{F}_2^{15}$ 中构造出大小为 192 的西顿集，而此前已知的最大集合大小为 152。使用逆函数与 Dobbertin 函数同样能构造出比以往已知更大的西顿集。每个在 $\mathbb{F}_2^t$ 中的新大西顿集 $M$ 都对应一个具有 $t$ 个校验位、最小距离为 5 且长度前所未见的二进制线性码。此外，我们改进了任意 APN 函数线性度的上界。