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?