Regular two-graphs from the even unimodular lattice \(E_ 8\oplus E_ 8\) (Q1357264)

A two-graph is a three-regular hypergraph \((V,E)\) with the additional property that each four-element subset of \(V\) contains an even number of edges from \(E\). A two-graph is regular if each two-element subset of \(V\) is contained in the same number of edges. Each regular two-graph is associated with a set of equiangular lines in some Euclidean space whose dimension depends on the two-graph. The author studies those regular two-graphs whose sets of equiangular lines can be obtained by some complicated process from vectors of the lattice \(E_8 \oplus E_8\), as well as Steiner triple systems and other structures related to these two-graphs.