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

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