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 ${\mathbb{F}}_q$-linear sets contained in a line of $\text{PG}(2,q^t)$. We show that scattered ${\mathbb{F}}_q$-linear sets of rank $3$ in $\text{PG}(1,q^3)$ correspond to particular hyperbolic quadrics and that ${\mathbb{F}}_q$-linear clubs in $\text{PG}(1,q^t)$ 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 $\text{PG}(5,q)$.