A tower of geometries related to the ternary Golay codes

The Steiner system ∑ = S(12,6,5) admits a unique lax projective embedding / in PG(V), V = F(6,3). The embedding / induces a full projective embedding e of the dual Δ of ∑ in the dual PG(V*) of PG(V). The affine expansion Afe(Δ) of Δ to AG(V*) (also called linear representation of Δ in AG(V*)) is a flag-transitive geometry with diagram and orders as follows: Its collinearity graph is the minimal distance graph of the 6-dimensional ternary Golay code. We shall prove that Afe(Δ) is the unique flag- transitive geometry with diagrams and orders as above. The {0,1, 2,3,41- residues of Afe(Δ) can also be obtained as affine expansions from the dual of S(11,5,4) and are related to the 5-dimensional ternary Golay code. We shall characterize them too by their diagram and orders. Finally, the {0,1,2,3}-residues of Afe(Δ) are isomorphic to the affine expansion of the dual of the classical inversive plane of order 3. A characterization will also be given for these expansions, in the same style as for Afe(Δ).