Some new results on small blocking semiovals

A blocking semioval S in a projective plane of order q is a set of points such that every line meets S in at least one point (the blocking property), and every point of S lies on a unique tangent line (the semioval property). The set of points on the sides of a triangle, excluding the vertices, is a blocking semioval in any projective plane of order q > 2, and has size 3q − 3. The size of a blocking semioval is constrained to be between 2q + 2 and q √ q + 1, and generically we call a blocking semioval “small” if its size is less than 3q − 3. In this paper we give several new constructions of small blocking semiovals in even order Desarguesian planes and also prove some nonexistence results in planes of order 8 and 9 which allow us to determine exactly the size of the smallest blocking semioval in most planes of these orders. In addition, we give an improvement to the lower bound on the size of a blocking semioval in a plane of order q to a value on the order of 2q + √ 2q.