On large Sidon sets

TL;DR

This paper introduces a novel method for constructing large Sidon sets in binary vector spaces using APN functions with high linearity, resulting in larger sets than previously known and new binary linear codes.

Contribution

It demonstrates that APN functions with high linearity can be used to construct larger Sidon sets, improving known bounds and generating new codes.

Findings

Constructed Sidon sets of size 192 in _2^{15}

Achieved larger Sidon sets using inverse and Dobbertin functions

Improved upper bounds for the linearity of APN functions

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.