A study of $k$-arcs in Hughes plane of order 9 by means of computer

Using the method of step-by-step identifications, as well as a computer, authors study $k$-arcs of Hughes plane of order 9 for $k=3, 4,\dots, 10$. The following results are obtained. For $k=6,7,8$, and 9 there are 1720, 1427, 24, and 4 types of incomplete $k$-arcs, respectively. There are 198 types of complete 7-arcs and 316 types of complete 8-arcs. For $k=3,4,5$ and for some types of 6-arcs, our results repeat the earlier ones of V. I. Vasil'kov. The results concerning complete $k$-arcs for $k=6,9,10$ coincide with those of Denniston (1971). For $k$-arcs of each type, the group of automorphisms is described and the total number of aires is found which are isomorphic to a given arc with respect to the group of all collineations in Hughes plane of order 9.