Abstract
The proper linear spaces on 17 points are classified. The computation is based on the parameters of the geometries and makes extensive use of tactical decompositions. A specific one, the tactical decomposition by ordering (TDO) which has been invented by Betten and Braun (Combinatorics’90, Ann. Discrete Math., Vol. 52, North-Holland, Amsterdam, 1992, pp. 37–43) is presented in full detail. The TDO may be seen as the final step of parameters of the geometries. In the current article, the authors show how the TDO can be used in order to construct geometries. This new method starts by calculating all possible TDO-schemes which the requested geometries may have. In a second step, all geometries for a fixed TDO-scheme are constructed. This two-step approach is a versatile tool which may be applied to other construction problems, too. The current work may be seen as an extension of A. Betten and D. Betten (J. Combin. Des. 7 (1999) 119–142) where all linear spaces on 12 points were classified.
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