The parity-check matrix of a non-binary low-density parity-check (LDPC) code over GF(q) is constructed by assigning non-zero elements from GF(q) to the 1s in that of the corresponding binary LDPC code. In this paper, we provide a theorem that establishes a necessary and sufficient condition that a q-ary matrix constructed by assigning non-zero elements from GF(q) to the 1s in the parity-check matrix of a binary quasi-cyclic LDPC code must satisfy in order for its null-space to define a non-binary quasicyclic LDPC (NB-QC-LDPC) code over GF(q). We then propose a general scheme for constructing NB-QC-LDPC codes along with some other code construction schemes that might serve better for different design goals. We also demonstrate that NBQC-LDPC codes are very suitable for high-rate applications, e.g. applications in magnetic recording and storage systems and optical communication systems.