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We describe a construction of primitive 2-designs and strongly regular graphs from the simple groups , and . The designs and the graphs are constructed by defining incidence structures on conjugacy classes of maximal subgroups of , and . In addition, from the group , we construct 2-designs with parameters and having the full automorphism group isomorphic to .

We introduce the notion of q-analogs of strongly regular graphs and give some first examples of such structures. We prove a necessary condition on the parameters and show the connection to designs over finite fields.

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A construction of designs acted on by simple primitive groups is used to find some 1-designs and associated self-orthogonal, decomposable and irreducible codes that admit the simple group He of Held as an automorphism group. The properties of the codes are given and links with modular representation theory are established. Further, we introduce a method of constructing self-orthogonal binary codes from orbit matrices of weakly self-orthogonal designs. Furthermore, from the support designs of the obtained self-orthogonal codes we construct strongly regular graphs with param