The simple incidence structure formed by points and unordered pairs of distinct parallel lines of a finite affine plane of order is a design. If , is the complementary design of . If , is isomorphic to the geometric design (see [2; Theorem 1.2]). In this paper we give necessary and sufficient conditions for a design to be of the form for some finite affine plane of order . As a consequence we obtain a characterization of small designs .
A combinatorial characterization of finite projective planes using strongly canonical forms of incidence matrices is presented. The corresponding constructions are applied to known projective planes of order 9. As a result a new description of the Hughes plane of order nine is obtained.
The paper has been presented at the International Conference Pioneers of Bulgarian Mathematics, Dedicated to Nikola Obreshkoff and Lubomir Tschakalo ff , Sofia, July, 2006.Two heuristic algorithms (M65 and M52) for finding respectively unitals and maximal arcs in projective planes of order 16 are described. The exact algorithms based on exhaustive search are impractical because of the combinatorial explosion (huge number of combinations to be checked). Algorithms M65 and M52 use unions of orbits...