The linear spaces on $\le$ 12 points


The number of linear spaces on $v$ points LIN$(v)$ 
is {\sc Sloane}'s Sequence no. A001200~\cite{SLOANE} 
(encyclopedia number M0726~\cite{SLOANEWWW}): 
$1,1,1,2,3,5,10,24,69,384,5250,232929$
for $v = 0, 1, \ldots, 11$. \\
It is the aim of this talk to extend this sequence 
by the term ${\rm LIN}(12) = 28872973.$ \\
The construction of geometries has a preprocessing phase where 
possible parameters of the geometries are computed algebraically. 
So, the geometries are enumerated according to their line type. 
If the subcases by line type are too big for construction 
one specifies the point types and the distribution of points of different type. 
If these subcases by point types are still too large, refined 
line cases are introduced. \\
This method of varying depth of parameter precomputation allows to 
react in a flexible way to different grades of difficulty 
in the construction of linear spaces. \\
The TDO-method of {\sc Betten} and {\sc Braun}~\cite{BETTENBRAUN} is used 
as a tool for classification of the geometries. The isomorphism problem 
is solved by using canonical forms of the geometries. The canonical form 
respects the TDO invariant and can be computed efficiently if 
the geometry already has a good decomposition by its TDO. \\
AMS subject classification: 05B25, 05B30, 51E99


\bibitem{BETTENBRAUN}
\mbox{{\BF Dieter Betten, Mathias Braun:}} 
A tactical decomposition for incidence structures. 
{\em Ann. Disc. Math.} {\bf 52} (1992), 37-43. 

\bibitem{SLOANE}
\mbox{{\BF Neal J.A. Sloane:}} 
A handbook of integer sequences. 
{\em Academic Press} 1973.

\bibitem{SLOANEWWW}
\mbox{{\BF Neal J.A. Sloane:}} 
Database of integer sequences (WWW version) \\
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