Non-transversal Vectors of Some Finite Geometries

By means of associated structural invariants, we efficiently construct four biplanes of order 9 - except the one with the smallest automorphism group, that is found by Janko and Trung. The notion of non-transversal vector is introduced since we observed related properties that provide significantly more efficient constructions. There is a dichotomy in the structure of biplanes of order 7 and 9 with respect to the incidence matrix symmetry.