On large Sidon sets
Ingo Czerwinski, Alexander Pott
TL;DR
The paper tackles the problem of constructing large Sidon sets in $\mathbb{F}_2^{t}$ by exploiting the link between APN (almost perfect nonlinear) functions and Sidon sets via the graph $G_F$. By intersecting Sidon sets with affine hyperplanes and leveraging the linearity $\mathcal{L}(F)$ of APN functions, it derives a method to obtain Sidon sets in $\mathbb{F}_2^{2n-1}$ of size $2^{n-1}+\mathcal{L}(F)/2$, and uses high-linearity APN examples (notably certain eight-variable APN functions, the inverse function, and the Dobbertin family) to produce record Sidon sets, including a $192$-element Sidon set in $\mathbb{F}_2^{15}$. These constructions translate to binary linear codes with $t$ check bits and minimum distance at least 5, yielding new code lengths (e.g., $[191,176,5]$) and infinite families. The work also tightens upper bounds on APN linearity and highlights open questions about possible infinite families achieving larger Sidon sets, linking combinatorial Sidon-type constructions with cryptographic function properties and coding theory.
Abstract
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.