On some 10-arcs for deriving the minimum order for complete arcs in small projective planes Original Research Article

In this paper, we provide a group-theoretic computer-free construction of some 10-arcs in PG(2,q) which have the dihedral group of order 6 as stabiliser and which have similar geometrical properties to some complete 10-arcs of minimum size in PG(2,q), for q=17,23. Having at oneʹs disposal such 10-arcs in PG(2,q), for almost all q, we analyse, by means of a computer search, some possibilities of completing them and we prove that such 10-arcs give a way of determining several examples of very small complete arcs in PG(2,q). For q=17,23,43,49,53,61,79,101,103,107,113,121,127 we determine examples of arcs with size smaller than or equal to the known complete arcs.