Mixed partitions and related designs
We define a mixed partition of pi = PG(d, q(r)) to be a partition of the points of pi into subspaces of two distinct types; for instance, a partition of PG(2n - 1, q(2)) 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 q(2n). Here we show that our partitions can be used to construct generalized Andre planes, thereby providing a geometric representation of an infinite family of generalized Andre 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(r(2)n - 1, q) and hence produce 2-(q(r2n), q(rn), 1) ( translation) designs with parallelism. These designs are not isomorphic to the designs obtained from the points and lines of AG(r, q(rn)).
mixed partitions spreads generalized andre planes singer groups translation designs