  The Wigner function provides a useful quasiprobability representation of
quantum mechanics, with applications in various branches of physics. Many nice
properties of the Wigner function are intimately connected with the high
symmetry of the underlying operator basis composed of phase point operators:
any pair of phase point operators can be transformed to any other pair by a
unitary symmetry transformation. We prove that, in the discrete scenario, this
permutation symmetry is equivalent to the symmetry group being a unitary
2-design. Such a highly symmetric representation can only appear in odd prime
power dimensions besides dimensions 2 and 8. It suffices to single out a unique
discrete Wigner function among all possible quasiprobability representations.
In the course of our study, we show that this discrete Wigner function is
uniquely determined by Clifford covariance, while no Wigner function is
Clifford covariant in any even prime power dimension.
