New Upper Bounds on the Sizes of Caps in $PG(N,5)$ and $PG(N,7)$

Abstract

Let $m_2(N,q)$ denote the size of the largest caps in $PG(N,q)$ and let $m_2\prime (N,q)$ denote the size of the second-largest complete caps in $PG(N,q)$. Presently, it is known that $m_2(4,5)\le 111$ and that $m_2(4,7)\le 316$. Via computer searches for caps in $PG(4,5)$ using the result of Abatangelo, Larato, and Korchmáros that $m_2\prime (3,5)=20$, we improve the first upper bound to $m_2(4,5)\le 88$. Computer searches in $PG(3,7)$ show that $m_2\prime (3,7)=32$, and this latter result then improves the upper bound on $m_2(4,7)$ to $m_2(4,7)\le 238$. We also present the known upper bounds on $m_2(N,5)$ and $m_2(N,7)$ for $N>4$.