Some Results Concerning {(q + 1)(n - 1); n}-Arcs and {(q + 1)(n - 1) + 1; n}-Arcs in Fintie Projective Planes of Order q

In a finite projective plane zr(q) of order q, any nonvoid set of k points may be described as a {k; n}-arc, where n(n % 0) is the greatest number of collinear points in the set. {k; 2}-arcs may be more simply called k-arcs. In a given plane a {k; n}-arc is said to be complete if there exists no {k'; n}-arc, k ' ~ k , which contains it. For given q and n(n v ~ 0), k can never exceed (n -1)(q -k 1) -k 1, and an arc with that number of points will be called a maximal arc [1]. Equivalently, a maximal arc may be defined as a nonvoid set of points meeting every line in just n points or in none at all. Evidently a maximal arc is complete. We remark that 7r(q) is a maximal {q2 _? q _]_ 1; q -k1)-arc and that ~r(q)\line L is a maximal {q2; q}-arc. I f K is a {qn q + n; n}-arc (i.e., a maximal arc) of a projective plane 7r(q) of order q, where n ~ q, then it is easy to prove that the set K ' = {lines L of zr(q)11L n K ---;~} is a {q(q n + 1)/n; q/n}-arc O.e., a maximal arc) of the dual projective plane 7r*(q) of 7r(q). It follows immediately that, if the desarguesian projective plane PG(2, q) over the Galois field GF(q) contains a {qn q + n; n}-arc, n ~ q, then it also contains a {q(q n + 1)/n; q/n}-arc. From the preceding it follows that a necessary condition for the existence of a maximal arc (as a proper subset of a given plane 7r(q)) is that n should be a factor of q. But the condition is not sufficient; Cossu [2] has proved that, in the desarguesian plane of order 9, there is no {21; 3}-arc. In [3], Denniston proves that the condition does suffice in the case of any desarguesian plane of order 2 h. Recently the author [5] has proved that