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An algorithm for constructing minimal TANT networks is presented. Using the upper prime permissible implicants as candidates, two particular networks, A and B, are constructed. Network A has the minimum number of third level gates among all networks that have the minimum number of second level gates. Network B has the minimum number of second level gates among all networks that have the minimum number of third level gates. For most functions with few variables, it can be shown that one or both of these two networks is a minimal TANT network. (No function with less than nine variables has been found which violates the above statement.) For other functions, upper and lower bounds for both the second level and third level gates are derived. These bounds provide extra constraints which can speed up the process for finding a minimal TANT network.