Sidon sets and 2-caps in $\mathbb{F}_3^n$

TL;DR

This paper introduces the concept of d-caps in finite vector spaces over GF(3), proves that 2-caps are equivalent to Sidon sets, and investigates the maximum size of 2-caps in these spaces.

Contribution

It generalizes the notion of caps in GF(3)^n, establishes the equivalence between 2-caps and Sidon sets, and explores their maximal sizes.

Findings

2-caps in GF(3)^n are exactly Sidon sets

Characterization of 2-caps in finite vector spaces over GF(3)

Analysis of the maximum size of 2-caps in GF(3)^n

Abstract

For each natural number $d$, we introduce the concept of a $d$-cap in $\mathbb{F}_3^n$. A subset of $\mathbb{F}_3^n$ is called a $d$-cap if, for each $k = 1, 2, \dots, d$, no $k+2$ of the points lie on a $k$-dimensional flat. This generalizes the notion of a cap in $\mathbb{F}_3^n$. We prove that the $2$-caps in $\mathbb{F}_3^n$ are exactly the Sidon sets in $\mathbb{F}_3^n$ and study the problem of determining the size of the largest $2$-cap in $\mathbb{F}_3^n$.