A probabilistic construction of small complete caps in projective spaces

TL;DR

This paper introduces a probabilistic method to construct small complete caps in projective spaces, achieving bounds close to theoretical limits and improving previous results for various dimensions.

Contribution

It presents a new probabilistic construction technique that significantly reduces the size of complete caps in projective spaces, approaching the lower bounds.

Findings

Constructed complete caps of size $O(q^{(N-1)/2} \, \log^{300} q)$

Improved upper bounds for minimal length of certain covering codes

Close asymptotic bounds to the trivial lower bound

Abstract

In this work complete caps in $PG(N,q)$ of size $O(q^{\frac{N-1}{2}}\log^{300} q)$ are obtained by probabilistic methods. This gives an upper bound asymptotically very close to the trivial lower bound $\sqrt{2}q^{\frac{N-1}{2}}$ and it improves the best known bound in the literature for small complete caps in projective spaces of any dimension. The result obtained in the paper also gives a new upper bound for $l(m,2,q)_4$, that is the minimal length $n$ for which there exists an $[n,n-m, 4]_q2$ covering code with given $m$ and $q$.