This work aims to jointly optimize the coding and node selection to minimize the processing time for distributed computing tasks over wireless edge networks. Since the joint optimization problem formulation is NP-hard and nonlinear, we leverage the discrete characteristic of its decision variables to transform the problem into an equivalent linear formulation. This linearization can guarantee to find the optimal solutions and significantly reduce the problem's complexity. Simulations based on real-world datasets show that the proposed approach can reduce the total processing time up to 2.3 times compared with that of state-of-the-art approach.