A higgledy-piggledy set of planes based on the ABB-representation of linear sets

In this paper, we investigate the André/Bruck-Bose representation of certain Fq-linear sets contained in a line of PG(2,qt). We show that scattered Fq-linear sets of rank 3 in PG(1,q3) correspond to particular hyperbolic quadrics and that Fq-linear clubs in PG(1,qt) are linked to subspaces of a certain 2-design based on normal rational curves; this design extends the notion of a circumscribed bundle of conics. Finally, we use these results to construct optimal higgledy-piggledy sets of planes in PG(5,q).