A 35-set of type (2, 5) in PG(2, 9)

If z has order q* and n is of the form n =m+q, then either k= m(q*+q+ l), or k= (m+q)(q*-q+ 1). Obviously, in case m=O, K is a maximal arc. Moreover, if m = 1 and q is a prime power, then K is either a Baer subplane or a unital [lo]. Furthermore, m pairwise disjoint Baer subplanes of n always yield a set of type (m, m + q) and size m(q* + q + 1). In case rc is Desarguesian, such a set does exist as the plane has a partition into Baer subplanes [3,9]. On the other hand, it was proved in [6] (see also [7]) that any Hughes plane contains a set with the same parameters as the union of m mutually disjoint Baer subplanes which does not split into m Baer subplanes. Now we turn to sets of type (m, m+q), m>2, and size k= (m + q)(q* q + 1). Not many of such sets are presently known. We list those we are aware of. A construction is given in [4] when m = (q 1)’ provided 71 contains a subplane of order q 1. Under the assumptions q* is a fourth power and the plane is Desarguesian, a set with the considered 303 0097-3165/87 $3.00