Finite Projective Planes and Orthogonal Arrays

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We got the following four results.(1)It was proved that there is no projective plane of order 12 admitting a collineation group of order 16 by showing the nonexistence of some orthogonal array OA (72,12,6,2). We used a computer for our research.(2)It was proved that there is only one symmetric transversal design STD_4[12;3] up to isomorphism. We also showed 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.(3)We constructed a symmetric transversal sesign STD_7[21;3] admitting an automorphism group of order 7 which acts semiregularly on the set of the point groups and on the set of block groups.(4)Let S be a blocking semioval in arbitrary projective plane Pi of order 9 which meets some line in 8 points. According to Dover in [2], 20leqvert Svertleq 24. In [7] one of the authors showed that if Pi is desarguesian, then 22leqvert Svertleq 24. In this note all blocking semiovals with this property in all non-desarguesian projective plane of order 9 are completely determined. In any non-desarguesian plane Pi it is shown that 21leqvert Svertleq 24 and for each iin{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.