On Complete Caps in the Projective Geometries over ${\mathbb{F}}_3$ II: New Improvements

Abstract

Hill, Landjev, Jones, Storme, and Barat proved in a previous article on caps in $PG(5,3)$ and $PG(6,3)$ that every 53-cap in $PG(5,3)$ is contained in the 56-cap of Hill and that there exist complete 48-caps in $PG(5,3)$. The first result was used to lower the upper bound on $m_2(6,3)$ on the size of caps in $PG(6,3)$ from 164 to 154. Presently, the known upper bound on $m_2(6,3)$ is 148. In this article, using computer searches, we prove that every 49-cap in $PG(5,3)$ is contained in a 56-cap, and that every 48-cap, having a 20-hyperplane with at most 8-solids, is also contained in a 56-cap. Computer searches for caps in $PG(6,3)$ which use the computer results of $PG(5,3)$ then lower the upper bound on $m_2(6,3)$ to $m_2(6,3)\le 136$. So now we know that $112\le m_2(6,3)\le 136$.