  The construction of optimal line packings in real or complex Euclidean spaces
has shown to be a tantalizingly difficult task, because it includes the problem
of finding maximal sets of equiangular lines. In the regime where equiangular
lines are not possible, some optimal packings are known, for example, those
achieving the orthoplex bound related to maximal sets of mutually unbiased
bases. In this paper, we investigate the packing of subspaces instead of lines
and determine the implications of maximality in this context. We leverage the
existence of real or complex maximal mutually unbiased bases with a
combinatorial design strategy in order to find optimal subspace packings that
achieve the orthoplex bound. We also show that maximal sets of mutually
unbiased bases convert between coordinate projections associated with certain
balanced incomplete block designs and Grassmannian 2-designs. Examples of
maximal orthoplectic fusion frames already appeared in the works by Shor,
Sloane and by Zauner. They are realized in dimensions that are a power of four
in the real case or a power of two in the complex case.

Functional Analysis;Combinatorics