On Caps and Cap Partitions of Galois Projective Spaces

Let PG(N, q) be the projective space of dimension N over the finite field GF (q). A k–cap K in PG(N, q) is a set of k points, no three of which are collinear [16], and a k–cap is called complete if it is maximal with respect to set–theoretic inclusion. The maximum value of k for which there exists a k–cap in PG(N, q) is denoted by m2(N, q). This number m2(N, q) is only known, for arbitrary q, when N ∈ {2, 3}. Namely, m2(2, q) = q + 1 if q is odd, m2(2, q) = q + 2 if q is even, and m2(3, q) = q 2 + 1, q > 2. With respect to the other values of m2(N, q), apart from m2(N, 2) = 2 N , m2(4, 3) = 20, m2(5, 3) = 56, only upper bounds are known. Finding the exact value for m2(N, q), N ≥ 4 and constructing an m2(N, q)–cap seems to be a very hard problem. Partitions of projective spaces into caps have recently received some attention. In this direction, B.C. Kestenband [18] and later on G.L. Ebert [8] adopting a different method, proved that the projective space PG(2n, q) can be partitioned into caps of size (q + 1)/(q + 1). Also, G.L. Ebert [8] proved that PG(2n − 1, q), n even can be partitioned into caps of size q+1. Moreover, these partitions are neverthless “uniform”, in the sense that the objects have all the same geometric nature. Sometimes it is impossible to partition a given projective geometry uniformly (on arithmetic grounds) although a uniform partition may become possible if one or more objects of a given kind are removed. We shall call such a non–uniform partition a mixed partition of a projective geometry. B.C. Kestenband [19] proved the existence of a mixed partition of PG(2n − 1, q) consisting of two (n − 1)–subspaces and q − 1 caps of size (q − 1)/(q − 1). Also, using the process of “lifting ” a collineation of PG(2, q) to a collineation of PG(5, q) preserving a quadric Veronesean, it was shown in [3] the existence of a mixed partition of PG(5, q) consisting of two planes and q − 1 quadric Veroneseans. In this paper we will study some “special” caps and cap–partitions (mixed partitions) from a group–theretic point of view. The idea is that if a configuration of points in a projective space is “special”, for instance a cap is