  Motivated by the work of Kalloniatis, Pauli and Pinsky, we consider the
theory of light-cone quantized $QCD_{1+1}$ on a spatial circle with periodic
and anti-periodic boundary conditions on the gluon and quark fields
respectively. This approach is based on Discretized Light-Cone Quantization
(DLCQ). We investigate the canonical structures of the theory. We show that the
traditional light-cone gauge $A_- = 0$ is not available and the zero mode (ZM)
is a dynamical field, which might contribute to the vacuum structure
nontrivially. We construct the full ground state of the system and obtain the
Schr\"{o}dinger equation for ZM in a certain approximation. The results
obtained here are compared to those of Kalloniatis et al. in a specific
coupling region.
