A note on constructing digraphs with prescribed properties

Let n be a non-negative integer and k be a positive integer. A digraph D is said to have property Q(n, k) if every subset of n vertices of D is dominated by at least k other vertices. For q == 5 (mod 8) a prime power, we define the quadruple Payley digraph D~4) as follows. The vertices of D~4) are the elements of the finite field F q' Vertex a is joined to vertex b by an arc if and only if a b = y4 for some y E F q' In this paper, we show that for sufficiently large q, the digraph D~4) has property Q(n, k).