Regular Two-graphs from the Even Unimodular Lattice E8⊕E8

Starting from the even unimodular latticeE8⊕E8,one constructs odd systems (i.e. sets of vectors with odd inner products) of 546 vectors using results of Deza and Grishukhin. One studies the subsystems consisting of 36 pairs of opposite vectors spanning equiangular lines. These subsystems represent regular two-graphs. This gives 100 such two-graphs and they coincide with the first 100 in a list of 227 two-graphs generated by E. Spence. Using the root systems of the sublattices generated by the 100 odd systems, the set of the 100 two-graphs is divided into seven classes. The first four classes correspond to the 23 Steiner triple system on 15 points containing a head, i.e. a Fano plane.