Some directed strongly regular graphs constructed from linear groups

Abstract In this paper, we classify directed strongly regular graphs admitting a transitive action of one of the linear groups $$L_2(q), q\le 32$$ , $$L_3(q), q \le 7$$ , and $$L_4(2)$$ , for which the rank of the permutation representation is at most 20. Several of the graphs found have a so far unknown parameter set. Some examples generalize to an infinite family.