  Measurements of the $^2$H($^{16}$C,$^{16}$C$^{*}$$\rightarrow^4$He+$^{12}$Be
or $^6$He+$^{10}$Be)$^2$H inelastic excitation and cluster-decay reactions have
been carried out at a beam energy of about 23.5 MeV/u. A specially designed
detection system, including one multi-layer silicon-strip telescope at around
zero degrees, has allowed the high-efficiency three-fold coincident detection
and therefore the event-by-event determination of the energy of the unstable
nucleus beam. The decay paths from the $^{16}$C resonances to various states of
the final $^{10}$Be or $^{12}$Be nucleus are recognized thanks to the
well-resolved $Q$-value spectra. The reconstructed resonances at 16.5(1),
17.3(2), 19.4(1) and 21.6(2) MeV are assigned as the $0^+$, $2^+$, $4^+$ and
$6^+$ members, respectively, of the positive-parity
$(3/2_\pi^-)^2(1/2_\sigma^-)^2$-bond linear-chain molecular band in $^{16}$C,
based on the angular correlation analysis for the 16.5 MeV state and the
excellent agreement of decay patterns between the measurements and theoretical
predictions. Moreover, another intriguing high-lying state was observed at
27.2(1) MeV which decays almost exclusively to the $\sim$6 MeV states of
$^{10}$Be, in line with the newly predicted pure $\sigma$-bond linear-chain
configuration.
