In this paper, we investigate the construction of a transmit signal for a base station with a massive number of antenna arrays under the cost-effective 1-bit digital-to-analog converters. Due to the coarse nonlinear property, conventional precoding methods could not yield an attractive performance with a severe error-floor problem. Moreover, finding an optimal transmit signal is computationally implausible because of its combinatorial nature. Thus, it is still an open problem to construct a 1-bit transmit signal efficiently. As an extension of our earlier work, we propose an efficient method to construct an 1-bit transmit-signal under quadrature-amplitude-modulation constellations. Toward this, we first derive the so-called feasibility condition which ensures that every user's noiseless observation belongs to a desired decision region, and then transform it as linear constraints. Taking into account the robustness to an additive noise, the proposed construction method is formulated as a well-defined mixed-integer-linear-programming problem. Based on this, we develop a low-complexity algorithm to solve it (equivalently, to generate a 1-bit transmit signal). Via simulations, we verify the superiority of the proposed method in terms of a computational complexity and detection performance.