Algorithms for Finding Unitals and Maximal Arcs in Projective Planes of Order 16

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 of di erent subgroups of the automorphism group of the 273x273 bipartite graph of the projective plane. Two very efficient algorithms (developed by the author and not described here) are used in M65 and M52: (i) algorithm VSEPARN for computing the generators, orbits and order of the graph automorphism group; (ii) graph isomorphism algorithm derived from VSEPARN. Four properties are proved and used to speed up the algorithms M65 and M52. The results of these algorithms are published. After changing only the parameters of these algorithms they can be used for determining unitals in projective planes of different orders.