  We construct minimum-uncertainty states and a non-negative quasi probability
distribution for quantum systems on a finite-dimensional space. We reexamine
the theorem of Massar and Spindel for the uncertainty relationof the two
unitary operators related by the discrete Fourier transformation. It is shown
that some assumptions in their proof can be justified by the use of the
Perron-Frobenius theorem. The minimum-uncertainty states are the ones that
saturate this uncertainty inequality. The continuum limit is closely analyzed
by introducing a scale factor in the limiting scheme. Using the
minimum-uncertainty states, we construct a non-negative quasi probability
distribution. Its marginal distributions are smeared out. However, we show that
this quasi probability is optimal in the sense that there does not exist a
non-negative quasi probability distribution with sharper marginal properties if
the translational covariance in the phase space is assumed. Generally, it is
desirable that the quasi probability is complete, i.e., it contains full
information of the state. We show that the obtained quasi probability is indeed
complete if the dimension of the state space is odd, whereas it is not if the
dimension is even.
