Transformation of designs and other incidence structures (Q1205440)

In the 1960's there was a great need for different examples of projective and affine planes. The reviewer [Trans. Am. Math. Soc. 111, 1-18 (1964; Zbl 0117.373)] discovered a process, called derivation, which yielded new planes from known planes of finite square order. Later, the reviewer generalized this process to one, called net replacement, which did not require square order. [See the reviewer, ``Finite Translation Planes'' (1970; Zbl 0205.499).](Although the reviewer describes net replacement for translation planes, it works for arbitrary planes in exactly the same way.) Recently, \textit{E. F. Assmus} jun. and \textit{J. D. Key} [J. Geom. 37. No. 1/2, 3-16 (1990; Zbl 0705.51011)] described another procedure for generating new planes from known (translation) planes. The authors of the present article describe a procedure which generalizes the reviewer's replacement method to incidence structures. Although the authors' examples are known examples of net replacement, their procedure is a natural extension of the reviewer's one and should give rise to some interesting incidence structures. The authors apply their procedure to show that all known sharply 3-transitive finite permutation sets containing the identity can be obtained from the group \(PGL(2,q)\).