I am aware of one construction technique, involving 8-dimensional subspaces of a 24-dimensioinal vector space to create the octads. This technique is shown in *12 Sporadic Groups*.


However. I am interested in a way of constructing them from the projective plane S(2,5,21).
It is a matter of combinatorics to show that this projective plane exists. But from this I want to construct *the* (is there more than one?) S(3,6,22) steiner system, by adding a point and so on until I get to S(5,8,24). What does this method involve?



How to construct S(5,8,24)
See <http://www.win.tue.nl/~hansc/eidmamathieu.pdf> for a description from an incidence geometric point of view.


