  In the context of topcolor-assisted techicolor($TC2$) models, we investigate
the feasibility of detecting the non-universal gauge boson $Z^{'}$ in the
future high-energy linear $e^{+}e^{-}$ collider($LC$) experiments by performing
$\chi^{2}$ analysis via studying its virtual effects on the processes
$e^{+}e^{-}\to f\bar{f}$ with $f=\tau$, $\mu$, b, and c. We find that the
non-universal gauge boson $Z^{'}$ is most sensitive to the process
$e^{+}e^{-}\to \tau\bar{\tau}$. Discovery limits of the $Z^{'}$ mass
$M_{Z^{'}}$ can be enhanced by the suitably polarized beams. If we assume that
polarization of electron beam and positron beam are 80% and 60%, the $LC$
experiment with $\sqrt{s}=500GeV$ and $\pounds_{int}=340fb^{-1}$ can explore
$M_{Z^{'}}$ up to 8$TeV$ for the coupling parameter $K_{1}\leq$0.8 at 95% $CL$.
