  We present a high-precision timing analysis of PSR J1933$-$6211, a
millisecond pulsar (MSP) with a 3.5-ms spin period and a white dwarf (WD)
companion, using data from the Parkes radio telescope. Since we have accurately
measured the polarization properties of this pulsar we have applied the matrix
template matching approach in which the times of arrival are measured using
full polarimetric information. We achieved a weighted root-mean-square timing
residuals (rms) of the timing residuals of 1.23 $\rm \mu s$, 15.5$\%$
improvement compared to the total intensity timing analysis. After studying the
scintillation properties of this pulsar we put constraints on the inclination
angle of the system. Based on these measurements and on $\chi^2$ mapping we put
a 2-$\sigma$ upper limit on the companion mass (0.44 M$_\odot$). Since this
mass limit cannot reveal the nature of the companion we further investigate the
possibility of the companion to be a He WD. Applying the orbital period-mass
relation for such WDs, we conclude that the mass of a He WD companion would be
about 0.26$\pm$0.01 M$_\odot$ which, combined with the measured mass function
and orbital inclination limits, would lead to a light pulsar mass $\leqslant$
1.0 M$_\odot$. This result seems unlikely based on current neutron star
formation models and we therefore conclude that PSR J1933$-$6211 most likely
has a CO WD companion, which allows for a solution with a more massive pulsar.
