Hyper-Reguli of Dimension 3

Here we restrict attention to hyper-reguli in translation planes of order qn for n = 3. As mentioned, when n = 3, Bruck [179] has shown that every hyper-regulus that sits in a Desarguesian affine plane is an Andre´ hyper-regulus. Furthermore, Pomareda [1108] proved that each Andre´ hyper-regulus of order q3 can have exactly two possible Andre´ replacements.