The limitations of nice mutually unbiased bases

Mutually unbiased bases of a Hilbert space can be constructed by partitioning
a unitary error basis. We consider this construction when the unitary error
basis is a nice error basis. We show that the number of resulting mutually
unbiased bases can be at most one plus the smallest prime power contained in
the dimension, and therefore that this construction cannot improve upon
previous approaches. We prove this by establishing a correspondence between
nice mutually unbiased bases and abelian subgroups of the index group of a nice
error basis and then bounding the number of such subgroups. This bound also has
implications for the construction of certain combinatorial objects called nets.