On complete caps in the projective geometries over F_3. II.

Hill, Landjev, Jones, Storme and Bar\'at 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 $m_2(6,3) \leq 136$. So now we know that $112 \leq m_2(6,3) \leq 136$.