The correlation properties of sequences form a focal point in the design of multiple access systems of communications. A popular choice for the set of sequences to deploy is the quasi-complementary sequence set. There is a growing body of literature that recognises the importance of quasi-complementary sequence sets. In this paper, using additive characters over finite fields, we propose five classes of asymptotically optimal quasi-complementary sequence sets, including periodic and aperiodic ones. In particular, the designed asymptotically optimal quasi-complementary sequence sets have new parameters and small alphabet sizes. The small alphabet size enhances their appeal for implementation.