  A complete set of mutually unbiased bases in a Hilbert space of dimension $d$
defines a set of $d+1$ orthogonal measurements. Relative to such a set, we
define a "MUB-balanced state" to be a pure state for which the list of
probabilities of the $d$ outcomes of one of these measurements is independent
of the choice of measurement, up to permutations. In this paper we explicitly
construct a MUB-balanced state for each prime power dimension $d$ for which $d
= 3$ (mod 4). These states have already been constructed by Appleby in
unpublished notes, but our presentation here is different in that both the
expression for the states themselves and the proof of MUB-balancedness are
given in terms of the discrete Wigner function, rather than the density matrix
or state vector. The discrete Wigner functions of these states are
"rotationally symmetric" in a sense roughly analogous to the rotational
symmetry of the energy eigenstates of a harmonic oscillator in the continuous
two-dimensional phase space. Upon converting the Wigner function to a density
matrix, we find that the states are expressible as real state vectors in the
standard basis. We observe numerically that when $d$ is large (and not a power
of 3), a histogram of the components of such a state vector appears to form a
semicircular distribution.
