Luby Transform code (LT code) is the first practical digital fountain code and has been widely used as basic components in many communication applications. The coding behavior of LT code is mainly decided by a probability distribution of codeword degrees. In order to customize a degree distribution for different purposes, multi-objective evolutionary algorithm is introduced to optimize degree distributions in this paper. Two critical performance indicators of LT code are considered in our experiments. Some applications hope to minimize the overhead of extra packets and some require to limit the computational cost of the coding system. To handle this problem, MOEA/D is applied to optimize two objectives simultaneously. We expect to obtain the Pareto front (PF) formed by partial optimal solutions and provide those available degree distributions to different LT code applications. Not only promising results are represented in this paper but also the behavior of LT code is thoroughly explored by optimizing the degree distribution according to multi-objectives.