The logic function synthesis using ULM U$_h$ is constructed based on canonic Reed-Muller expansion coefficient for a given function. This paper proposes the fast synthesis algorithm using ULM U$_h$ for ternary function. By using circuit cost and synthesis method of proposed in this paper, order of control input variable minimum number of ULM U$_h$ can be decided in the synthesis of n-variable ternary function. Accordingly, this method enables to optimum circuit realization for ternary function synthesis using ULM ULM U$_h$ and can be applied to ternary function synthesis using ULM U$_h$. The complexity of search for select the order of all control input variables is (n+2)(n-1)/2.