# On k-caps in PG(n, q), with q even and n \geq 3

## TL;DR

This paper improves bounds on the size of k-caps in projective geometries over finite fields with even q, specifically refining the second largest complete k-cap size in PG(3, q) and deriving new bounds for higher dimensions.

## Contribution

It provides improved upper bounds for the second largest complete k-cap in PG(3, q) and new bounds for maximum k-cap sizes in higher-dimensional projective spaces over even q.

## Key findings

- Improved upper bound for m'_2(3, q) when q ≥ 8 and q even.
- Derived new bounds for m_2(n, q) for n ≥ 4 and q even.
- Identified gaps in previous proofs by Cao and Ou.

## Abstract

Let m_2(n, q) be the maximum size of k for which there exists a k-cap in PG(n, q), and let m'_2(n, q) be the second largest value of k for which there exists a complete k-cap in PG(n, q). In this paper Chao's upper bound q^2 - q + 5 for m'_2(3, q), q even and q \geq 8, will be improved. As a corollary new bounds for m_2(n, q), q even, q\geq 8 and n \geq 4, are obtained. Cao and Ou published a better bound but there seems to be a gap in their proof.