On an infinite class of Steiner systems constructed from affine spaces
The authors show that a construction introduced by \textit{E. F. Assmus} and \textit{J. D. Key} [On an infinite class of Steiner systems with \(t=3\) and \(k=6\), J. Comb. Theory, Ser. A 42 (1986)] can be adapted to apply to the designs of points and lines of the finite affine spaces AG(d,q) for \(q\geq 3\). An infinite class of \(3-(q^ d+1,q+1,1)\) designs is constructed for \(d>2\), \(q\geq 3\), by extending the 2-design of points and lines of AG(d,q) in the following way: a new point ''\(\infty ''\) is adjoined to each line in AG(d,q), and for each plane AG(2,q) in AG(d,q), an arbitrary class of ovals in the plane is chosen so that the plane is extended to an inversive plane. In this way, new 3-designs are constructed that are shown not to be isomorphic to the Witt designs [see, for example, \textit{D. R. Hughes} and \textit{F. C. Piper}, Design Theory (1985; Zbl 0561.05009)]. The authors also note that the class of designs constructed is not the same as that constructed by \textit{H. Hanani} [J. Comb. Theory, Ser. A 26, 1-19 (1979; Zbl 0395.05007)] of the same parameters, although there is some overlap.
scientific article; zbMATH DE number 3935051