Large $(k; r, s; n, q)$-sets in Projective Spaces

TL;DR

This paper studies the maximum size of special point sets in projective spaces over finite fields, establishing asymptotic sizes for various parameters and generalizing classical bounds.

Contribution

It introduces new asymptotic bounds for the size of $(r,s)$-sets in projective spaces and constructs large examples for specific parameters.

Findings

Existence of $(3, 2)$-sets of size $(1+o(1)) q^{3/2}$ in $ ext{PG}(6,q)$

Existence of $(4, 2)$-sets of size $(1+o(1)) q^{(n-1)/2}$ for general $n$

Generalized Rao's bound to $O(q^{(n-e+1)/e})$ for certain $(r,s)$-sets

Abstract

A $(k; r, s; n, q)$-set (short: $(r,s)$-set) of $\mathrm{PG}(n, q)$ is a set of points $X$ with $|X| = k$ such that no $s$-space contains more than $r$ points of $X$. We investigate the asymptotic size of $(r, s)$-sets for $n$ fixed and $q \rightarrow \infty$. In particular, we show the existence of $(3, 2)$-sets of size $(1+o(1)) q^{3/2}$ for $n=6$, $(4, 2)$-sets of size $(1+o(1)) q^{\frac{n-1}{2}}$, and $(9, 2)$-sets of size $(1+o(1)) q^2$ for $n=4$. We also generalize a bound by Rao from 1947 and show that an $(r,s)$-set has size at most $O(q^{\frac{n-e+1}{e}})$ if there exist integers $d,e \geq 2$ such that $s=d(e-1)$ and $r=de-1$.