Directed strongly regular graphs with rank 5

From the parameters (n,k,t,λ,μ)(n,k,t,λ,μ) of a directed strongly regular graph (dsrg) A. Duval (1988) [4] showed how to compute the eigenvalues and multiplicities of the adjacency matrix, and thus the rank of the adjacency matrix. For every rational number q  , where 15≤q≤710, there is a feasible (i.e., satisfying Duval's conditions) parameter set for a dsrg with rank 5 and with kn=q.In this paper we show that there exist a dsrg with such a feasible parameter set only if kn is 15, 13, 25, 12, 35, or 23. Every dsrg with rank 5 therefore has parameters of a known graph. The proof is based on an enumeration of 5×55×5 matrices with entries in {0,1}{0,1}.