Blocking Semiovals of Type (1, m+1, n+1)

We consider the existence of blocking semiovals in nite projective planes which have intersection sizes 1; m + 1 or n + 1 with the lines of the plane, for 1 m < n. For those prime powers q 1024, in almost all cases, we are able to show that, apart from a trivial example, no such blocking semioval exists in a projective plane of order q. We are able to prove also, for general q, that if q 2 + q + 1 is a prime or three times a prime, then only the same trivial example can exist in a projective plane of order q. 1. Motivation. Blocking sets in projective planes have been much studied; the `classical' results due to Bruen 6] and 7] state that in a projective plane of order q, a blocking set has between q + p q + 1 and q 2 ? p q points. Many additional references, as well as descriptions of applications in game theory and cryptography, can be found in Chapter 8 of Batten 1]. A semioval in a projective plane is a set of points S such that for each point P of S, there exists a unique line which meets S in exactly the point P. In 13], Hubaut proved that in a projective plane of order q, a semioval S has between q + 1 and q p q + 1 points. These two extremes occur in the case when S is an oval (see 1]) or a unital (see 10]) respectively. In the case of regular semiovals, that is, when S has constant line size a, considered as a design in its own right, Blokhuis and Szz onyi 5] prove that either S is an oval or ajq ? 1. Blocking sets and semiovals coincide in the case when each is a unital. In fact, for minimal blocking sets (where each point is on at least one tangent), it is known that the upper bound on the number of points is q p q + 1, which is precisely the unital case (see 8]). This leads to the more general question: in what other cases is a blocking set also a semioval? There is a trivial example on 3(q ? 1) points in every nite projective plane of order q > 2. Take three non-copuntal lines, a \triangle", and delete the three points where …