Non rank 3 strongly regular graphs with the 5-vertex condition

Three new strongly regular graphs on 256, 120, and 135 vertices are described in this paper. They satisfy thet-vertex condition — in the sense of 1] — on the edges and on the nonedges fort=4 but they are not rank 3 graphs. The problem to search for any such graph was discussed on a folklore level several times and was fixed in 2]. Here the graph on 256 vertices satisfies even the 5-vertex condition, and has the graphs on 120 and 135 vertices as its subgraphs. The existence of these graphs was announced in 3] and 4]. 4] contains M. H. Klin's interpretation of the graph on 120 vertices. Further results concerning these graphs were obtained by A. E. Brouwer, cf. 5].