Some Friends of Alltop’s Designs \(4 -(2^f +1,5,5)\)

We construct several families of simple 4-designs, which are closely related to Alltop’s series with parameters \(4-(2^f+1,5,5)\), \(f\) odd. More precisely, for every \(q=2^f\), where \(gcd(f,6)=1\), \(f\geq5\), we construct designs with the following parameters:

\[4-(q+1,6,\lambda),\, \text{where}\, \lambda\in\{60,70,90,100,150,160\},\]
\[4-(q+1,8,35),\]
\[4-(q+1,9,\lambda),\, \text{where}\, \lambda\in\{63,147\}.\]