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This paper presents a method based on orthogonal arrays of constructing pure quaternary quantum error-correcting codes. As an application of the method, some infinite classes of quantum error-correcting codes with distances 2, 3, and 4 can be obtained. Moreover, the infinite class of quantum codes with distance 2 is optimal. The advantage of our method also lies in the fact that the quantum codes we obtain have less items for a basis quantum state than existing ones.