The design of optimum networks has for sometime been formulated as an integer programming problem; however, only recently have networks been obtained by using these formulations. This note reports some computational experience on the synthesis of optimum networks using Gomory's all-integer integer method. A typical example has a problem size of 91 unknowns and 240 inequalities. The effects of further restricting the problem by adding additional constraints are also investigated. Some other experiments, such as various ordering of inequalities and convergence characteristics of feasible versus infeasible problems, are described.