Mutual Unbias IV
Ericsson's construction considers the $d+1$ bases as a polytope defined by the convex hull of points lying on certain simplices (with $d$ vertices defining a basis) in $(d-1)$ dimensional planes in the quantum state space (or rather, a density operator space) of dimension $d^2 - 1$, which is to say the space of $d \times d$ Hermitian matrices of trace 1. For example, when $d=2$, the polytope is the octahedron on a Bloch sphere (didn't that polytope come up just the other day?).
The problem of fitting the regular simplices into the polytope is shown to be equivalent to finding $d-1$ orthogonal Latin squares! This is the same as finding a finite affine plane of order $d$.