LDPC codes based on multi-edge protographs potentially have larger minimum distances compared to their counterparts, single-edge protographs. However, considering different features of their Tanner graph, such as short cycles, girth and other graphical structures, is harder than for Tanner graphs from single-edge protographs. Here, we provide a novel approach to construct the parity-check matrix of an LDPC code which is based on trades obtained from block designs. We employ our method to construct multi-edge quasi-cyclic (QC) LDPC codes.We use those trade-based matrices to define base matrices of multi-edge protographs. The construction of exponent matrices corresponding to these base matrices has less complexity than the ones proposed in the literature. We prove that these base matrices result in QC-LDPC codes with smaller lower bounds on the lifting degree than existing ones.