Geodesic
The Translation Planes of Dempwolff
Canadian journal of mathematics, Tome 33 (1981) no. 5, pp. 1060-1073

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In [2], Dempwolff constructs three translation planes of order 16 using sharply 2-transitive sets of permutations in S 16. That is, if acting on Λ is a sharply 2-transitive set of permutations then an affine plane of order n may be defined as follows: The set of points = {(x, y)|x, y ∊ Λ} and the lines = {(x, y)|y = xg for fixed }, {(x, y)|x = c}, {(x, y)|y = c} for c ∊ Λ.Let V be a vector space of dimension k over F ≅ GF(pr). A translation plane may be defined by finding a set M of pkr – 1 linear transformations such that xy –l is fixed point free on for all x ≠ y in M.Notice that if we allow V to act on itself then MV is a sharply 2-transitive set on V if and only if xy –1 is fixed point free on for all x ≠ y in M. 
Johnson, N. L. The Translation Planes of Dempwolff. Canadian journal of mathematics, Tome 33 (1981) no. 5, pp. 1060-1073. doi: 10.4153/CJM-1981-081-8
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[2] 2. Dempwolff, V., Einige Translationsebenen der Ordnung 16 und Ihre Kollineationen. (To appear.) Google Scholar

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