Mixed partitions and related designs

We define a mixed partition of Π = PG(d, qr) to be a partition of the points of Π into subspaces of two distinct types, for instance, a partition of PG(2n− 1, q2) into (n− 1)-spaces and Baer subspaces of dimension 2n− 1. In this paper we provide a group theoretic method for constructing a robust class of such partitions. It is known that a mixed partition of PG(2n−1, q2) can be used to construct a (2n−1)spread of PG(4n − 1, q) and, hence, a translation plane of order q2n. Here we show that our partitions can be used to construct generalized Andrè planes, thereby providing a geometric representation of an infinite family of generalized Andrè planes. The results are then extended to produce mixed partitions of PG(rn− 1, qr) for r ≥ 3, which lift to (rn − 1)-spreads of PG(r2n − 1, q) and hence produce 2− (qr2n, qrn, 1) (translation) designs with parallelism. These designs are not isomorphic to the designs obtained from the points and lines of AG(r, qrn).