Abstract:In this article, we prove that there does not exist a symmetric transversal design STD 2 [12; 6] which admits an automorphism group of order 4 acting semiregularly on the point set and the block set. We use an orbit theorem for symmetric transversal designs to prove our result. As a corollary of the result, we prove that there is no projective plane of order 12 admitting a collineation group of order 8.

Let S be a blocking semioval in an arbitrary projective plane Π of order 9 which meets some line in 8 points. According to Dover in [2], 20 ≤ |S| ≤ 24. In [8] one of the authors showed that if Π is desarguesian, then 22 ≤ |S| ≤ 24. In this note all blocking semiovals with this property in all non-desarguesian projective planes of order 9 are completely determined. In any non-desarguesian plane Π it is shown that 21 ≤ |S| ≤ 24 and for each i ∈ {21, 22, 23, 24} there exist blocking semiovals of size i which meet some line in 8 points. Therefore, the Dover's bound is not sharp.

In this note we construct a symmetric transversal design ST D 7 [21; 3]. Therefore there exists an orthogonal array OA (63, 21,3,2).

In this article we prove that there is only one symmetric transversal design STD 4 [12; 3] up to isomorphism. We also show that the order of the full automorphism group of STD 4 [12; 3] is 2 5 · 3 3 and Aut STD 4 [12; 3] has a regular subgroup as a permutation group on the point set. We used a computer for our research.

The study of blocking semiovals in finite projective planes was motivated by Batten [1] in connection with cryptography and was begun by Dover [4,5]. In this note, two new families of blocking semiovals are constructed in desarguesian planes.